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NOTES ON MECHANICAL DRAWING 


Introductory to Machine Design 


ARRANGED FOR STUDENTS IN 


Mechanical Engineering 


P. M. CHAMBERLAIN, M. E. 

PROFESSOR OF MECHANICAL ENGINEERING 

• IN 

LEWIS INSTITUTE 


SECOND EDITION 

CHICAGO, ILL. 


1 904 . 




I 


LIBRARY CONGRESS 
Two copies Received 

OCT 10 1304 

_ Copyright Entry 
CLASS O- XXo. No. 

Z/7«7 2- 

' COPY B 



Copyrighted 1934 
by P. M. Chamberlain. 







Leyda Bros., Printers, 
794 Madison Street, 
Chicago. 


TMPS6-024376 






PREFACE 

* 


The first edition of this collection of notes printed three years ago 
served its purpose to a measure, and it is hoped that the revisions and 
additions will make it more nearly meet the requirements of students in 
drawing in the earlier years of an engineering course. The additions 
are brief treatments of 1st and 3rd angle projections; intersections of 
planes and solids, solids with solids, developement of solids, revolution 
of solids, location of points of conic sections and practical considera¬ 
tions involved in the construction of simple spur and bevel gearing. 

There is appended some exercises which may be used but which 
are intended as examples of classes of exercises which are preferably 
arranged from time to time by the instructor. The use of trade cata¬ 
logs in giving general proportions of various machines and details is 
highly satisfactory and develops the student’s sense of proportion as 
dimensioned parts or problems worked out by formulae can never do. 

Suggestions from Messrs. Mosely, DePuy, Hatch, Parker, Hawley, 
Bacon, Schenck and Barnay, who- have given instruction at periods or 
continuously along the line of these notes, have been incorporated and 
material has been taken from the general literature bearing on the sub¬ 
ject. Where the matter is unique, acknowledgement is made in the text. 

Chicago September, 1904. 





NOTES ON MECHANICAL DRAWING. 


In any drawing-room it may be observed that the instruments act¬ 
ually used are very few in number. It is therefore advised that the 
beginner start with only those most needed and, if possible, the best ob¬ 
tainable, which are American or American Type Swiss instruments. 

1 he celluiod triangles are preferable to all others on account of 
their transparency and cleanliness. One 30°x60 c x90 Q about 8 inches 
long and one 45 x45 c x90° about 4 inches long are convenient sizes. 

The X square should be slightly longer than the drawing board 
and without swivel head. One of good quality pear wood is quite satis¬ 
factory but more elaborate ones can be had. The head of the T square 
is apt to swell from a true surface where it slides along the edge of the 
board. This may be corrected by occasionally scraping away the bowed 
part. 

The compasses should be the large size supplied with pencil and 
pen legs and adjustable needle point. Large dividers usually come 
with sets but are not very necessary. A set of bow instruments; bow 
dividers, bow pen and bow pencil, complete the necessary tools. The 
last two instruments may be had combined in one instrument whose 
needle point remains stationary while the pen or pencil leg is twirled 
around it. 



This instrument, Fig. 1, well made, is, for small circles, preferable 
to the old type of bow instruments. 

The pen usually called the right line pen should be about six inches 
long and with spring blade but no hinge, as shown in Fig. 2. 



The catalogs of makers of, and dealers in, drawing instruments 
offer considerable information regarding the construction and care of 
instruments. 

Higgins’ water-proof ink is a standard liquid ink but usually is too 
thin when fresh. Evaporation soon brings it to the right consistency. 
Any ink will dry up at the pen point and clog the flow, hence it is nec¬ 
essary to wipe the pen out frequently with a piece of linen. Scraps of 
tracing cloth washed free from all starch make good pen wipers It is 
not necessary to change the adjustment of the screw in the pen to clean 
it properly. 


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The scale should be boxwood, steel, or boxwood covered with 
white celluloid. For machine drawings the proportions usual are 3 in. 
= 1 ft., whence £ in. = 1 in.; in. = 1 ft., whence J in. = 1 in. ; or f 
i n * — 1 ft., whence y 1 ^ in. = 1 in. These divisions are readily found on 
any rule. The triangular architect’s scale has these and other divisions, 
and is convenient to have, but troublesome to use, owing to the fact 
that the scale divisions wanted are usually the last found. A four-foot 
folding rule with white background is a most convenient thing to have 
in the pocket and is accurate enough for detail drawings which are to be 
traced. 

For ordinary pencilling the pencil 
should have a long tapering screw¬ 
driver point, as shown in Fig. 3. 

The hardness of the pencil should be 
such as will enable the user to make 
with rapid strokes clear lines which 
can be erased without leaving depressions in the paper. Johann Faber’s 
HHHH is recommended for detail paper. 

The drawing-board should have one edge straight and should be 
free from warp and wind. The various constructions can be observed 
in dealers’ catalogs. 

For most detail work a white or buff drawing paper with a sharp 
surface is desirable. As pencil drawings are seldom inked-in, the inking 
qualities need not be considered. It is desirable to have sizes of draw= 
ings conform to some standard. Where large quantities are consumed 
the paper is usually bought in rolls and a set of dimensions which may 
easily be remembered, and which cut to advantage, is preferable. If 36 
inch paper is used, a standard of 24 inches by 36 inches is desirable. 
This may be doubled or divided into 2, 4, or 8 parts tor smaller sizes, as 
shown in Figure 4. 

Inside of the dimensions 
thus formed always lay off 
a border line f of an inch 
from the edge. In making 
a tracing it is desirable to 
allow the cloth extra mar¬ 
gin outside of the full di¬ 
mensions of the standard 
as the edges of the tracing 
become curled with much 
handling. The line of full 
size should be drawn to 



indicate on the blue print where to trim. 



7 
































Standards for center lines, dimension lines and witness lines are 

found in some variety. Red ink lines which print faintly are sometimes 
used for each, but fine black lines with distinctive characteristics are for 
_ many reasons preferable. A dot and 



CENTRE 

and a dash for center lines, Fig. 5,two 


DIMENSION 

dots and a dash for dimension lines, 


several dots and a full line for wit¬ 
ness lines should be used in draw¬ 
ings made in connection with these 
notes. Arrow heads should be large enough to indicate readily to what 
the dimension refejs and may be made free hand, the width 
being about one-third the height. When the space for a dimension 
is too small to admit the arrow heads and dimension, the arrow heads 
may be placed pointing toward each other and the dimension placed 
outside. Never place the dimensions any great distance away from the 
place it would be looked for. 

In making a tracing it is usually best to ink-in first the small 
curves, then the larger ones, and then the straight lines, as it is easier to 
join a large curve or a straight line to a small curve than vice versa. The 
neatness of a tracing depends to a great extent on getting good tangen- 
cies. For those who cannot judge the proper junction with their eye it 
is advisible to draw a pencil line through the centers of the two curves, 
as shown in Figure 6, thus determining where they shall come together. 
In joining a curve to a straight line, the center used should lie on a 
perpendicular from the end. In joining a straight line to a curve, it 
should form a right angle with a radial line drawn to point of contact. 

In tracing, the picture lines should be nearly ^ of. an inch wide 
and cross section, witness, centre, dimension, and dotted lines as light 
as can be neatly made. Arrow heads should be distinct and quite acute. 
All lettering and dimensioning should be distinct and uniform in size. 
The drawing pen should be used for making figures and letters. The 
glossy side of the tracing cloth should be used, and well chalked before 
inking. 

In case of necessity of erasure, use a piece of talc crayon to restore 
gloss to the cloth. Never use a knife to erase as the ink will soak into 
the cloth where this is done. A tracing should not be commenced until 
the pencil drawing is complete and checked. 


HIDDEN 


WITNESS 



PICTURE 

FIG. 5. 




FIG. 6. 


8 
























The making of compound curves is usually done with an “irregular 
curve,” a scroll with a great variety of curves and seldom the one de¬ 
sired. The more satisfactory and quite as rapid method is to sketch in 
free-hand the desired curve, or locate points if a known curve, and then 
locate sufficient centers to draw the curve with the compasses. See arc 
methods, pages 21 to 26. See also Figure 0. 

In all drawings of cast parts there should be shown a fillet where- 
ever there are two parts coming together at an angle. These should be 
made with a bow pen or bow pencil. 

Cylindrical pieces are shown in section only when there is internal 
arrangements requiring it. All cylindrical surfaces and circles should 
have center lines indicated. The material to be used in construction 
may be indicated by some convention in cross-sectioning, but economy 
of time and clearness of expression are better served by using plain 
hatching for all materials, and indicating by note, word, or initial the 
material. Where several pieces in contact are shown in section, the 
hatch lines may contrast in direction or spacing if necessary, but always 
at forty-five degrees. Lines used in hatching should ordinarily be 
about y'g of an inch apart and as fine as the pen will properly make. 
There are many ingenious devices to facilitate equal spacing, but it is 
usually preferable to space by eye. See Figure 5. 

When a piece is to be machined or finished and it is not entirely 
obvious that such is the case, it 

is usual to place two letters Jr J 1 

along the projection of the finished surface. When patts are to be fin¬ 
ished all over they may be marked finish all ovef*. 

The location of a dimension should always be such as will readily 
catch the eye of the mechanic and avoid all probable chance of con¬ 
fusion. The study of how a piece should be made in the shop is indis¬ 
pensable to good judgment in this very important consideration. This 
supposes that a designer is capable of directing every step of con¬ 
struction of that which he designs, as indeed he must be, to produce 
good designs and clear drawings. To be complete, a drawing must 
contain all dimensions and answer all questions that might arise to pat¬ 
tern maker, blacksmith, or machinist. Dimensions of more than 24 
inches are usually expressed in feet and inches thus: 38" should be 
written 3'—2". 

In lettering, the letter used should be one that can be quickly 
made, neat, and very easily read. For ease of acquirement and sim¬ 
plicity, the gothic letter, so called by printers, is satisfactory. Where a 
large letter is required, the following letter will be found useful for 
making precise, uniformly spaced letters. See Figure V. Lay off a line 
for the top limit of the legend and beneath it a point at a distance equal 
to the desired height; divide this space into five equal parts by the eye. 




The first trial will 
likely be unsuccess¬ 
ful, but a few trials 
should bring success 
Through these points, 
or short lines, draw 
lines parallel to the 
top one. The ac¬ 
curacy of the spacing 
may be tested by lay¬ 
ing diagonally over 
the lines any five 
equal spaces, on the scale, larger than the spacing on the paper. 
With the 45 degrees triangle lay off oblique lines as shown. Take care 
that the angular lines meet exactly at the top and bottom lines. Allow¬ 
ing seven spaces for all letters excepting 1, one space between letters 
and four between words, calculate the total number of spaces needed, 
and from the middle of the space allotted for the words, count to the 
left half the total spaces and begin laying off the upright lines of 
the letters at the intersection of the oblique with the parallel 
lines. This forms the outline of block letters which can easily 



be supplied with curves, the same centers serving for the inside and 
outside curves. On page 11 is given the construction of all letters and 
figures of this scheme, and the same plan of spacing may be applied to the 
five space letter, page 9. The word “letters” Fig. 8 is given in three sizes 
having exactly the same proportion, the smallest exhibiting approximately 
what should be used for free-hand letters, which are made with the right 
line pen held upright and so adjusted that the width of the line is the 
same whichever direction the pen is moved, the pen not being turned 
from the position used for drawing horizontal lines. In inking the let¬ 
ters, certain parts are made advantageously in a certain sequence and 


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CHJKLM 

NQPQRS 

TUVWXY 



56789 - 

FIG. 9. 


in a certain direction. The numbered arrows in Fig. 10 indicate a 
desirable order. A standard height of 3-32" for all ordinary notes and 
dimensions is found desirable. 


S & 1 & 2 85 51 2 & 0 


FIG. 10 


When the dimension of a radius is given it should be followed by 
the word radius or abreviation rad. The arrow head should be 
placed at the outer end only, and if it is not evident where the center is, 
a very small circle may indicate it. The dividing line in a fraction 
should always be horizontal. Dimensions should read in the direction 
of the dimension line and from the bottom or right hand side of the 
sheet. The bottom is taken as one of the long dimensions of the sheet. 

All parts shown threaded or tapped are understood to be the U. S. 
Standard for the size shown unless otherwise stated. The screw threads 
known as V shape, Sellers’ or U. S. Standard, square and Powell 
will be here considered. The U. S. Standard is perhaps the most widely 
used, although the V thread is frequently found in commercial bolts, set 


12 





screws and machine screws in the smaller sizes. Where the screw is 
used to transmit power, the square thread or the Powell thread shoulp 
be used. The Powell thread has the advantage that the lash caused by 
wear may be corrected by closing up the nut.' 

The V thread has an angle of 60 
degrees between sides of the thread. 

The depth of the thread is .866 of 
the pitch. See Fig. 11. 

FIG. 11 



The Sellers’ thread has an angle 
of 60 degrees between threads. The 
top of the thread and the throat of the 
space are flattened one-eighth of the 
pitch. The depth is .6495 of the pitch. 

See Fig. 12. The pitch may be gotten 
approximately by the formula P—(.24X 1 D-J-.625)—.175. 

The square thread has a profile consistent with its name. The 
depth is one-half of the pitch. 



The Powell thread has an an¬ 
gle of 29 degrees between the sides 
of the threads. The depth of thread 
is one-half of the pitch. The flat is 
.3707 of the pitch. See Fig. 13. 


h-P-H 

^•37Q7P 7 ^ 


r\ i \ i x rm 


POWELLr 


FIG. 13 


The term pitch means the distance the screw would twist longitudin¬ 
ally during one rotation, but is more frequently used as the distance from 
one point on a thread seen in profile to the corresponding point of the 
adjoining one. In the case of single thread the definitions would agree 
but if the thread is double the screw would move a greater distance than 
the point to point distance. The term will be used in these notes as in¬ 
dicated in Figures 11, 12, and 13. 

If there are two or more independent threads the terms double screw 
thread, triple screw thread, etc., are applied. Figure 14 shows a double 
right hand, and a single left hand Sellers thread. Figure 15 shows a 
double left hand V screw thread and a single right hand Powell screw 
thread. Figure 16, shows a double right hand square screw thread with 
the helix connecting the profiles. See also Figure 48. It will be noticed 
that in the single screw thread the crown of the thread is opposite a de¬ 
pression, while in the double screw thread the crown is opposite a crown. 
Figures 14 and 15 are conventional to the extent of the lines connecting 
the profiles being made straight instead of helical. 













In preparing to draw any style of thread, first lay out the limiting 
•side lines and center line. Draw light cross lines at a distance apart 
■equal to one half of the pitch. 


For the Sellers’ thread, divide one of these spaces into four parts 
with the aid of the triangle, as in Fig. 17, or by the eye, and draw 


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FIG. 17 


across the figure a 
line through the 
proper point, which 
is one-eighth of the 
pitch. With the 30 
degree triangle draw 
the outlines of one 
side of a thread 
through the intersec¬ 
tion of the last drawn 
cross line and the 
limiting side line; this 
will fix the position 
of the root lines. 


Points are thus fixed by intersections for all the threads. 


14 





















































































To draw the Powell thread, lay 
off cross lines as before and root 
lines which may easily be located 
by aid of the 45 degree triangle. 
See Figure 18. With the two tri¬ 
angles obtain 15 degrees, this be¬ 
ing near enough to the actual 14£ 
degrees. Draw through the limit¬ 
ing lines, and the cross lines all 
the sides of threads corresponding 
to this position. In a similar way 
draw the other sides through the 
intersections of the cross lines and 
the root lines. 




Correct Incorrect 

FIG. 19 



Correct Incorrect 
FIG. 20 



Correct 
FIG. 21 


It is seldom necessary to detail threads as certain conventional 
threads are used. The conventions shown in Figures 19, 20 and 21 
may be used respectively for sizes as in the cuts Figures 19 and 20 show 
two incorrect forms which are to be avoided. The slant should be one- 
half the pitch as shown in the correct forms. 

Sizes of U. S. Standard threads, nuts and bolt heads are given with 
other data on page 51. 

Bolts and nuts may be drawn conventionally as in Figure 22. The 
end view of the nut drawn with a chamfer circle one and one-half times 
the diameter of the bolt + T y' gives the tangencies for the hexagon and 
projected gives the point at which the chamfer begins. The broad face 
curve has a radius of the diameter of the bolt and the small curves are 
drawn to meet it. Bolts are usually threaded about two and a half times 
the diameter. 


15 






























































-^3cr 


FIG. 22 


Set screws usual¬ 
ly have heads equal 
in diameter and thick¬ 
ness to the diameter 
of the screw. The 
various forms of 
points are shown in 
Figure 23. 







Vi'^q’V « ' 

CUP 15 OVAL - 
ROINT N POINT 


GJ ^ 



DOG 

POINT 


PLAT 

POINT 

FIG. 23 


CONICAL. HEADLESS 
POINT 


There seems to be no generally adapted standard for the proportions 
of the points. The cup point may be made one-half the diameter, the 
oval point struck with a radius equal to the diameter, the dog point of 
diameter three-fourths and length three-eights the screw diameter, the 
flat point and headless of end diameter three quarters the diameter of the 
screw and the conical point as shown in cut. The necks should be 
deeper than the threads and of length a little greater than the pitch. In 
drawing conventional threads the pitch is spaced by eye as in hatching 
without exact regard to the true pitch. 

Cap screws vary in proportion by different makers, but there is 
given on Page 52 a table compiled from the Chicago Screw Company’s 
catalog. Up to 1" diameter they are threaded f of the length under the 
head, except the flat head which is threaded to the head. 






FIG. 24 


16 






































































































































































































































In indicating a threaded hole use one of the conventions shown in 
Fig. 15, which are simple and clear. In cases where the screw is not 
shown, or where only an end view is given it should be marked 
tap 2," etc. 

Where the material is shown in section as in connection with the 
■£" bolt in Figure 24, the threading of the casting is indicated. 

Where it is desired to indicate that the casting is not threaded, a clear- 

« 

ance shown between screw and casting accomplishes it. 

When duplicate parts are required, mark the drawings 2 or 

THIS or 3 OF THIS, Or MAKE 2 Or MAKE 3, etc. 

When a taper is required the drawing should be marked with the 
taper per foot of length; thus a taper of f" per foot means a total 
decrease of diameter or width of f' f for each foot of length, and in that 
proportion for shorter lengths. When a taper is determined by given 
dimensions at each end of a given length, the approximate taper per 
foot should be given and marked appr ox, xarer. 

For securing pulleys to shafts various forms of keys have been 
used. The Sellers standard and the Woodruff system are tabulated on 
page 56. 

For the shanks of drills, lathe centers, etc., the standard tapers 
are customary. The Morse taper and the Brown and Sharpe taper are 
on page 56. 

Tapered pins for keying various pieces together are usually made 
one-quarter inch per foot. 

For laying off an angle of definite amount or measuring an angle 
without protractor, see page 46. 

Give evrry special piece on a drawing a number consisting of the 
number of the drawing where it first appears and some letter. In the 
case of cast parts, mark rax. no. with the drawing number and dis¬ 
tinguishing letter 

The title, drawing number, date, etc., should be in the lower right 
hand corner, lengthwise of the sheet. See Page 6. The number should 
also be placed in the upper left hand corner upside down so that in look¬ 
ing over a pile of drawings for a certain number, it can be readily found, 
even though some of them may be upside down. The title should ordi¬ 
narily state first the firm name; second, the machine or structure to 
which the drawing pertains; third, the distinctive title of the particular 
drawing; fourth, the date; fifth, the scale; sixth, the serial number of 
the drawing In addition to these items the name of the student and 
the hours charged to the drawing are added in the title used at Lewis 
Institute, shown in Figure 25. 


17 


STUDENT DRAWING 
LEWIS INSTITUTE 


NS SCALE = 

HRS. DR’G HRS. TR’C 

□ATS 

NAME 

FIG. 25 

Those parts of a title which occur on every drawing should 
be of a standard style and size. This may be obtained by having the 
tracings printed on a press, by using a rubber stamp, or by having a 
copy which can be copied or traced A rubber stamp does not give an 
impression that will print well and' should be filled in with ink if used 
on a tracing. 

The outlines of figures resulting from the intersection of cones and 
and planes, or conic sections, are frequently employed by the engineer. 
Based on mathematical relationship, graphic methods have been 
ingeniously devised for drawing them when various data are given. 
Several methods for determining points on these curves are here given. 
The ellipse, the most frequently used conic section, is a curve any point 
on which has two measurments to two points, called foci, the sum of 
which is always equal to the sum of corresponding measurements for 
any other point on the curve. The names given to the methods are 
more or less arbitrary but serve for convenient reference. 

String Method. Given the 
long diameter A B and the short 
diameter C D, Fig. 26. Take 
the half major axis A O as a 
radius R, and from C as a cen¬ 
ter, describe an arc cutting the 
major axis at F and F\ Set 
steel needles at F and F' and 
attach to them the ends of a fig”. 26 

fine thread whose length is equal to the long diameter A B. 
Using a pencil sharpened with a groove around the lead and very near 
the point, draw the ellipse using the thread loop to constrain the 
movement of the pencil point. 

This method is likely to give inaccuracies owing to the stretch of 
the string. 


c 



18 



















Radii Method. Given 
the long diameter A B and 
the short diameter C D, Fig. 
27. Locate foci F F' as in 
|b preceding problem. Lay off 
E A equal to A F. With 
F' as a center draw with any 
radius R an arc cutting the 
major axis at G. With F as 
a center draw an intersecting arc with. Radius R' equal to E G. The in¬ 
tersection H will be a point on the ellipse. Locate other points by 
similar process in the same quadrant. Complete the ellipse by arc 
method. 

This method leaves the points near the end somewhat in doubt owing 
to the small angle between the arcs. 



FIG. 27 


Trammel Method. Given 
the long diameter A B and 
short diameter C D, Fig. 28. 

On a piece of cardboard lay 
off H E equal to the half long 
diameter A B. Also lay off 
H G equal to the half short 
diameter C D. Moving the 
trammel so that G always 
coincides with the major axis and E with the minor axis, the point H 
will travel along the path of the ellipse. Locate points in one quadrant 
and complete the ellipse by arc method. 

The difficulty of keeping the points G and E exactly on the axis 
usually leads to errors. 

Machines for drawing ellipses are made on this principle. 

Revolution Method. Given the ma¬ 
jor axis A" B" and the minor axis C" D" 
Fig. 29. Conceive the ellipse as being the 
projection of a circle revolved about a di¬ 
ameter A" B" to the position C" D". 
Draw a circle, E" G" F" of diameter equal 
to major axis of ellipse and its horizon¬ 
tal plane projection E' F'. Lay off C" D" 
the minor axis of the ellipse. Project D" 
to the horizontal plane and with A' as a 
center and A' F' as a radius draw an arc 
cutting D'. Draw C' D'. Project any 
point G" to G\ revolve it to FT and project 
it down to a horizontal line passing through 
G". The intersection H" will be a point 



c 



FIG. 28 


FIG. 29 


19 








































on the ellipse. Locate several points in one quadrant and draw ellipse 
by arc method. 


The Circle Method. 

Given the two half diameters 
A O and C O, Fig. 30. Draw 
quadrants of two circles with 
r adii equal to the half diam¬ 
eters of the required ellipse. 

Draw any radial line as O H. 

From the points where it in¬ 
tersects the arcs draw lines 
parallel to the axes as H K 
and L K. Their intersection K will give a point on the ellipse. 
Locate points in one quadrant and complete ellipse by the arc method. 
This method with reasonable care gives good results. 



Rectangular Parallel¬ 
ogram Method. Given the 
parallelogram E F G H, Fig. 
31. Draw diameters A B and 
i-b C D. Divide A O into equal 
parts arid E A into the same 
number of equal parts. From 
C draw lines cutting the 
points on E A and from D 
draw lines cutting the points 
on A G. Counting from A the respective lin s will intersect at 
points on the ellipse. In similar manner divisions may be made on G D 
and O D with centers taken at B and D the extremities of the diameters. 
Draw ellipse by arc me hod. 

Oblique Parallelogram e 
Method. Given oblique par¬ 
allelogram E F G H, Fig. 32. 

Draw conjugate axes A B-C 
D. Divide C E into a num¬ 
ber of equal parts and simil¬ 
arly D H, C O, and O D. 

From B draw radial lines 
through the division on C O 
and O D and from A draw radial lines to the divisions on C E and 
D H. Starting at A the respective lines will intersect at points on the 
ellipse. Through the center O and the points draw lines on which set 
off points equidistant from O in the opposite direction. With O as a 












































center and radius O D or O C draw an arc cutting the path of the ellipse 
which can be determined for the vicinity by locating several points; 
through the intersection K, draw a line D K. Through O draw rectan¬ 
gular axes L M and N P perpendicular and parallel, respectively, to D K. 
Draw ellipse by arc method. 


Arc Method. For ap¬ 
proximating the curve through 
known points. Given the long 
and short diameters of an 
ellipse and points on one 
quadrant, Fig. 33. Find by 
trial a radius R with center E 
on major axis whose arc will 
cut several points on ellipse 
near extremity of long diam¬ 
eter. Find by trial a radius 
R" with center E" on minor 
axis extended whose arc will 
cut several points on ellipse 
near extremity of short diam¬ 
eter. Find by trial a radius with center E' whose arc will come 
tangent to the smaller arc externally and the greater arc internally and 
cut the remaining points of the ellipse Through E' and E draw a line 
extending it to the arcs at T and similarly through E' and E" to T'. 
These will be the points of tangency. Locate these centers and tangent 
points symmetrically for the other three quadrants and complete the 
ellipse with arcs, stopping them at the tangent points. By increasing 
the number of arcs, very close approximations may be had. 

Foci Method. To draw 
a tangent to an ellipse at 
any point L, Fig. 34. With 
half the long diameter as a 
radius and center at D, strike 
arc cutting major axis at F 
and F' the foci. From F and 
F' draw through the point L 
the lines F G and F' E. Bi¬ 
sect the angle G L E and 
draw T T perpendicular to 
the bisector H L and through the point L, then T T will be tangent to 
the ellipse at the point L and H L will be a normal. 




21 




























Instantaneous Center Method. For drawing a normal to an 
ellipse.. With N as a center, Fig. 31, and radius R equal to the half 
major axis, draw an arc cutting the minor axis at M. Draw M N and 
at the points of intersection with the axes at S and M, erect perpen¬ 
diculars M P and S P. Through their intersection at P draw a line P N 
which will be normal to the ellipse at N and a perpendicular to it 
through N would be tangential to the curve. 

The line M N corresponds to H D in the trammel method, Fig. 28. 
The end M if it moves would move along C O and for the instant can be 
considered as having a center on M P at any distance from M. The 
point S would move along AO and for the instant has a center anywhere 
on the line PS. As S and M are one piece with N they would all move 
together and M and S would have their instantaneous center at the inter¬ 
section of the perpendiculars or at P. The instantaneous center of al\ 
parts of N M is at P, hence with Pas a center and radius P N, the arc 
would for the instant coincide with the ellipse. 

The parabola, next to the ellipse the most used conic section, is a 
curve any point on which is always equally distant from a line called the 
directrix and a focal point. For obtaining points on a parabola we 
have the following methods: 



String Method. Given direc¬ 
trix CAD, axis A B and focus F 
Fig. 35. Tie a fine string to a 
triangle and make a loop at the 
end of the string which will reach 
to G. Put loop over pin at F 
and with slender pointed pencil 
keep string close to triangle as at 
H. Moving the triangle along the 
T square the pencil will trace a 
parabola. 


Compass Method. Given direc¬ 
trix C D axis A B focus F and ver¬ 
tex V, Fig. 36. Draw any line G H 
paralell to directrix, and with radius 
R equal to H A, describe arc from 
center F cutting line G H. The in¬ 
tersection I will give point on par¬ 
abola. Locate other points and draw 
curve by arc method. 



22 
























Isosceles Angle Method. 
Given axis A B vertex V and point 
C through which parabola is to 
pass, Fig. 37. Draw C D perpen¬ 
dicular to A B and make isosceles 
triangle with A V equal to V B. 
Draw E G parallel to C D and 
divide A C and A D into equal 
spaces. Connect corresponding 
points of divisions in opposite di¬ 
rections. To find point of tan- 
gency to any lines as 2.2. Lay off 
V H equal to V K and through H draw line parallel to E G The line 
will cut 2-2 at point of tangency and locate point on parabola. Draw 
curve by arc method. 



Paralellogram Method. 

Given axis V B vertex V and 
point M through which parabola 
is to pass, Fig. 38. Construct 
the paralellogram V A M B. 

Divide B M into any number of 
equal parts and M A into the 
same number of equal parts. 

From the points on B M drav 
lines parallel to the axis and through the points on A M lines converging 
to V. The intersection of the lines adjacent to B and A respectively 
will give a point on the parabola and similarly with each succeeding pair 
of lines. Draw curve by arc method. 






PARABOLA 


Arc Method For approxi¬ 
mating the curve through known 
points. Given the axis and points 
on the parabola Fig. 39. With 
center E on axis draw an arc which 
will include several points near 
the vertex, with a longer radius 
draw tangent to the first, an arc 
which will include more points. 
Draw line through the centers E E' 
cutting the arcs giving point of tan¬ 
gency A. Continue the process 


FIG. 39 

* 

until one branch of the parabola is drawn, then locate symmetrical 
centers and points of tangency for the other branch. The centers can 


23 

























be located by trial and if necessary the direction of the center can be 
found by drawing a perpendicular to a tangent drawn at any point. 

Bisector Method of drawing a 
tangent to a parabola. Given the 
axis A B, directrix CD, focus F and 
point on the parabola, M Fig. 40. 

Draw M F from point M to focus and 
M K perpendicular trom point M to 
directrix. Bisect the angle F M K 
and the bisector will be tangent to 
the curve at point M. 

The hyperbola, perhaps less used than the other conic sections, is a 
curve any point on which has two measurments to two focal points, the 
difference of which is always equal to the difference of corresponding 
measurments for any other point on the curve. The following methods 
may be used: 

Radii Method. Given the foci 
FF', the major axis A B, Fig. 41. With 
F as a center and any radius R greater 
than F B, describe an arc. With F' as 
a center and radius R' equal to R minus 
A B, describe an arc cutting the for¬ 
mer. The intersection will be a point 
on the hyperbola. In similar manner 
locate other points and approximate 
curve by arc method. 




Asymptote Method. Given the asymptotes 
O M and O N and some point D on the hyperbola, 
Fig. 42. Draw any line C E through D and the 
asympto es. Lay off E P equal to C D and P will be 
a point on the curve Other points such as L may 
be found in the same way. The bisector O X of the 
angle MON will be an extension of the major axis. 
Approximate curve by arc method. 

Rectangle Method. Given reference lines 
O M and O N and a point C on the curve, Fig. 43. 
Through the point C draw C D parallel to O N and 
C E parallel to O M. Through O draw any line O G, 



24 























FIG. 43 


major axis. This method is much 
dicator cards. 


where it cut C D and C E as a 
G and H erect perpendiculars, 
and where they intersect as at 
K will be a point on the curve. 
To find the point O, having given 
the base line and any two points 
on the curve as L and Q, con¬ 
struct the paralellogram LPQR 
and through P and R draw a line 
which will intersect base at O. 
The bisection X O of the angle 
MON will coincide with the 
in applying the hyperbola to in- 



FIG. 44 


O and M. 
parallel to 


Tangent Method. Given a point 
E on the hyperbola and the major axis 
A B, Fig. 44. Describe a semicircle on, 
and equal in diameter to, the major 
axis Draw C D parallel to the major 
axis and tangent to the semicircle. Draw 
a line E H through E perpendicular to 
the minor axis and another E K parallel 
to it. With O as a center and O K as a 
radius, strike arc cutting C D as at L. 

Lay off H M equal to C L. Draw asymptote through 
Draw any line N R parallel to major axis. Draw G P 

minor axis. Lay off N R equal to O P and 
R will be a point on the hyperbola. Sim¬ 
ilarly locate other points and approximate 
the curve by the arc method. 

Parallelogram Method. Given the 
major axis A B and point C on the hyperbola, 
Fig. 45. Construct rectangle CD EH. Di¬ 
vide G C into equal parts and D C into the 
same number of equal parts. From A draw 
lines to the divisions on G C and from B 
to those on D C. Counting from C, the re¬ 
spective lines will intersect on the hyperbola. 
Draw hyperbola by arc method. 



Arc Method of approximating the curve through known points. 
Given sufficient points on a hyperbola and the extension of the 
major axis, Fig. 46. Find by trial a radius with center E on axis which 


25 




















































will cut points near the axis. 
From where the arc begins 
to leave the hyperbola as at 
A, draw a line through E ex¬ 
tended On this line locate 
a center as at E' for an arc 
which will cut some succeed¬ 
ing points. Where this arc 
begins to leave the hyperbola 
as at B, draw a line through 
E',etc. Locate symmetrically 
about the axis centers and 
points of tangency and com¬ 
plete the hyperbola. 



FIG. 46 




Foci Method of drawing a tan¬ 
gent. Given in Fig. 47, the foci t F', 
the major axis A B and the point E, 
on the curve, required the tangent 
through the point E. Draw E F' and 
E F and bisect the angle they make at 
E and the bisector will be tangent to the 
curve. 

Major Axis Method of drawing 
a tangent. Given in Fig. 47 the major 
and the point H on the curve required the tangent at the 
Draw the semicircle ALB on and equal in diameter to 
the major axis. Through the point H, drop a perpendicular H K to 
the major axis extended. Draw K L tangent to the semicircle and from 
its point of tangency L, drop the perpendicular L P to the major axis. 
Draw the line H P and it will be tangent to the curve at H. 



FIG. 47 


axis A B 
point H. 


In drawing the helix it is necessary to remember that the curve un¬ 
wound would be an inclined plane of height equal to the pitch, and base 
equal to the circumference, and that wrapped on the cylinder it rises along 
the axis equal distances for corresponding portions of the circumference. 
In Figure 48 A/' A", is the pitch and A' E' the diameter of the cylinder. 
The distance A 1 " A", is divided into any convenient number of equal parts 
and the circle C' D' C', is divided into the same number of angles. 
Starting from A'' when the point rises one-eighth A/' A", it travels at the 
same time one-eighth of the circumference or A' B\ The intersection 
B" will then be a point on the curve. It is necessary only to locate 
points on one-half the curve as at A" E" as the remainder is symmetrical 
with it. To draw the curve, points can be transferred with bow dividers 


20 














to a thin piece of wood and a templet 
whittled, or the curve can be approxi¬ 
mated by arcs, and a straight line for a 
portion between B" and D". Two radii 
are usually sufficient. The root of the 
thread would have a different helix from 
the crown although it rises the same 
distance and is gotten from the same 
divisions of height and angle but with a 
different diameter. 

The Iogarthanic spiral used in some 
classes of gear and cam work has the 
peculiarity that a tangent to the curve 



always makes the same angle with the radial line to that point. It 
has also the characteristic of any bisector AD, Figure 49, being a 
mean proportional between AC and AB. To pass a logarithamic 
spiral through two points as B and C with pole at A, lay off A' B' and 
A'C' equal to A B and AC and on them construct a semicircle C' D' 
B'. Erect a perpendicular at A' bisect the angle B AC and lay off A D 
equal to A' D'. 


Curves used for teeth of gears are usually involute or cycloidal. 
The involute may be drawn by unwinding a string or fine wire from 
about a cylinder, keeping it taut with a pencil in a loop at its free end 
and causing the pencil to mark as it unwinds and moves away from the 
stationary cylinder. The cycloidal curves may be drawn by attaching 
a marking point on the surface of a cylinder and rolling the cylinder 
inside or outside of another, allowing the marking point to trace its 
lines on paper or soft metallic surface such as zinc. If it rolls outside 
it traces epicycloidal and if inside hypocycloidal curves. The circle 
which rolls is called the generating circle and that on which it rolls, the 
pitch circle For locating points of these curves with drawing instru¬ 
ments the fo lowing methods are given. 

To obtain points on the involute curve by divider method. Given 
the radius R of the pitch circle and the angle of obliquity of the line of 
action (usually 15 J ) Fig. 50. Draw arc of pitch circle, radial line and 
line of action, passing through their intersection at A Draw base circle 
tangent to line of action. At the intersections of radial lines 1,2, 3, etc. 


27 



































with the base circle, draw perpen¬ 
dicular lines to them tangent to the 
circle. With bow spring dividers 
step off from 1, short distances 
along the base circle to a point 
near 6 and without removing the 
dividers, return along the tangent¬ 
ial line the same number of spaces 
to some point C which will be a 
point on the involute. Other points 
are obtained in a similar manner. 

Do not indent the paper with the 
dividers, but use a fine pointed 
pencil to locate the point C with a sharp cross mark. Care should be 
taken not to put any bending action on the legs of the dividers. It is 
not necessary to have the spacing come out exactly at the point of 
tangency as the error introduced by taking the point which the spacing 
gives nearest to it, is inappreciable. It is sometimes advantageous to 
change the setting of the dividers for new points on the curve avoiding 
thereby the error due to the divider point slipping into an indenture 
previously made. 

To obtain points on the epicycloidal and hypocycloidal curves by 
the divider method. Having given radius of curves of pitch circle R 
and radius of generating circle R' Fig. 51, to obtain points on the 
epicycloid. Draw a portion of the pitch circle and the generating circle 
tangent to it at A. Strike an arc through the center of the generating 
circle concentric with the pitch circle. With centers on this arc draw 

arcs of the generating circle tangent to 
the pitch circle. Draw radial lines to 
locate points of tangency. With bow 
spring dividers, space off from A several 
steps along the pitch circle to some 
point near a tangency as at B and with¬ 
out removing the point of the dividers 
from the paper, step off the same num¬ 
ber of spaces along the generating circle 
as at C then C will be a point on the 
curve. Locate other points in a sim¬ 
ilar way. It is not necessary that the 
spaces shall come out exactly at the 
point of tangency as the error caused by stopping at a point near 
the tangent point for returning is inappreciable. Do not prick points 




28 















in the paper but have a very sharp pencil 
ready to make the final point on the gener¬ 
ating circle. Only with great care and a 
very sharp pencil will the results be satis¬ 
factory. 

For points on the hypocycloid, Fig. 
52, the same directions apply only that the 
generating circle rolls inside instead of 
outside. 

Points may be obtained on the cy¬ 
cloid by a method which will be called the 
Compass Method. For the epicycloid, 



FIG. 52 



Fig 53, step off with dividers or bow 
pencil equal spaces on generating and 
pitch circles from point of tangency. 
With radius R" equal to A7' strike arc 
from center taken at 7. With radius R" ' 
equal to 7-7' strick arc from a center at 
A. The intersection at C will be a 
point on the curve. Other points are 
obtained by the same procedure, using 
for center equally spaced distances on 
generating and pitch circles. 


FIG. 53 


The hypocycloid, Fig. 54, is treated 
•in a similar manner. This method has 
the advantage of giving the required points 
marked by the instruments in the process 
of construction, but in most hands is open 
to more likelihood of error than the pro¬ 
ceeding. 

Still another method somewhat similar 
to the last will be called the Arc Method 
to distinguish it 



For the epicycloid, Fig. 55, step off equal spaces on the pitch and 
generating circle. Draw an arc through the center of the generating 
circle from the center of the pitch circle and from this same center draw 
radial lines through some of the steps on the pitch circle intersecting 
the arc drawn through the center of the generating circle. With centers 
at these intersections draw arcs of the generating circle. From the 
















•center of the pitch circle draw an arc through some point 8' and whe re it 
cuts the generating circle, tangent at 8, as at C will be a point on the curve. 




With the generating circles drawn inside the pitch circle, Fig. 56, 
the hypocycloid may be determined in a similar manner. 

For obtaining points of a curve generated by a circle rolling on a 
straight line, usually called cycloid but more pi^perly orthocycloid, the 
methods given apply, remembering that the straighter line may be con¬ 
sidered the arc of a circle of such great radius that the radii became 
parallel. In the case of the involute it becomes a straight line making 
with the pitch line an angle corresponding to 90° minus 15° or whatever 
angle the system is based on. 




FIG. 57 


In representing 
any solid by a me¬ 
chanical drawing it is 
usually necessary to 
make by projection 
drawing two, three, 
or more views of it to 
bring various parts 
clearly before the eye. 


FIG. 58 

Two systems are in general use and they unfortunately are some¬ 
times confused. They are technically called first angle and third angle 
projection. Descriptive geometry discusses the subject fully, but the 
following explanation will cover the usual cases which come in the prac¬ 
tice of the mechanical draftsman. 


For the study of the first angle imagine a box with the three sides, 
nearest the eye, removed, the three remaining sides lined with drawing 
paper and the object to be represented suspended in the box as in Fig. 
57. The floor of the box corresponds to the horizontal plane or ground 
plan, the wall to the left the vertical plane or front elevation and the wall 


30 



















































to the right the perpendicular plane or side elevation. Looking directly 
down on the object all the lines visible are transferred or projected to 
the horizontal plane; looking at it from the right all the lines are pro¬ 
jected to the perpendicular plane, and so from the left, to the vertical plane. 
Spreading the paper flat would give the arrangement shown in Fig. 58. 

The third angle 
may be illustrated 
by imagining the 
nearest three sides 
of the box replaced 
with three sheets 
of transparent pa¬ 
per with the object 
suspended behind 
them. See Fig. 59. 

The sheets of 
transparent paper correspond to the horizontal, vertical, and perpendic¬ 
ular planes, and are frequently spoken of as top view, front view and right 
side view, respectively. Spreading the paper flat would give the arrange¬ 
ment shown in Fig. 60. The third 
angle method is the one usually em¬ 
ployed in machinery drawing. 

The derivation of the terms first 
angle, third angle, etc., may be seen 
from the arrangement of horizontal, 
vertical and perpendicular planes 
shown in Fig. 61. 

Sometimes it is desirable to show 
more than three views of an object, 
and the method consistent with the 




third angle projection 
is to put the top view 
above the front eleva¬ 
tion, the bottom view 
below it, the right el¬ 
evation to the right, 
and the back eleva¬ 
tion adjacent, either 
to the right or the left 
elevation. See Fig. 
62. Any arrange¬ 
ment can be used 
which is consistent 
with the idea of the 



31 

























































































































object being enclosed in a transparent box with the different views 
traced on the surfaces and then unfolded into one plane. 


The planes of pro- horizontal^^ 
jeccion thus far consid¬ 
ered have been at right 
angles to each other, but 


it is sometimes desir- 

VERTICAL ^ 

able to make oblique 

PLANE 


projections, and for this 

\ / 


purpose auxilliary 



planes are used. Take 

/ \ 

\ \ 

for example the intersec- 


'\ 

tion of a cone and a 


FIG. 







> 



plane in Fig. 63 Required the true outline on the plane. Construct 
an auxiliary plane parallel to the intersecting plane and project the 
intersection perpendicularly to it. The planes unfolded would be as 
seen in Fig. 64. 


Revolving the object is one method 
for getting oblique views when the view 
sought cannot be obtained by one pro¬ 
jection. The normals used for axes of 
p pn revolution are shown in Fig. 65 lettered 
HPN, VPN and PPN for horizon¬ 
tal plane normal, vertical plane nor¬ 
mal, and perpendicular plane normal 
respectively. If the object seen in Fig. 

66 were revolved around the H P normal, the horizontal projec¬ 
tion would not be changed in outline but would occupy a different 
relationship to the groundlines. The vertical and perpendicular 
projections would be changed in outline, but one very important 
thiog is to be noted: every point t is the same distance form the hori¬ 
zontal plane as it was before, hence the vertical heights can be 
projected directly. Let the problem be to revolve the obje t in Fig. 

66 about the H P normal so that its edges are 45 c with the horizontal 
ground line as in Fig 67; second, to revolve it from this position about 
the P P norm d so that its perpendicular pro-jection makes 30° with the 
horizontal ground line as in Fig 68; third, to revolve it from the third 
position about the V P normal so that the projection of one of its edges 
is parallel to the horizontal ground line as in Fig. 69. In Fig. 

67 the horizontal projection is copied exactly from Fig. 66, 
and the heights of the vertical projections are projected from the 
vertical elevation in Fig. 66. The perpendicular projection is obtained 
from its horizontal and vertical views. In Fig. 68 the PP pro- 



32 


















































F. G. 66 



FIG 67 


jection i s copied 
exactly from Fig. 

07, and the V P 
projection i s ob¬ 
tained by projec¬ 
ting the widths 
from the V P pro¬ 
jections in Fig. 67. 

The horizontal pro¬ 
jection is obtained 
from i t s vertical 
and perpendicular 
views. In Fig. 69 
the vertical view is 
copied exactly 
from Fig. 68. The 
horizontal view is 
found from the 
projection of new 
position of the FIG . 69 

vertical view and the horizontal view in Fig. 68. It is very necessary 
to keep clearly in the mind’s eye what actual changes have been wrought 
by the successive revolutions of the different views. It is sometimes 
necessary to number the various corners to assist in finding identical 
points projected from different views, but it is desirable to keep the 
proce s so well in mind that this is unnecessary. 







In exhibiting on a drawing the outline of intersection of two solids 
or of one solid and a plane, it is convenient to conceive the surfaces of 
the solids or planes as made up of lines or elements. For example, take 
a cylinder three inches long and two inches in diameter, it can be con¬ 
ceived as having been generated by a line three inches long revolving 


33 





















































































34 






























































about a longitudinal axis to. which it is parallel and one inch distant. If 
in a model the revolution were rapid enough the line would present to 
the eye a cylindrical surface. If the line halted every inch and a fine 
wire of the same length were substituted for it at each stop, the result 
would be seen from two views as in Fig. 70. If a plane made by a mov¬ 
ing straight line sidewise were to intersect the cylinder obliquely and the 
line developing the plane halted at such places as it came in contact 
with the selected elements of the cylinder, the appearance of one such 
intersection would be as pictured in Fig. 71. This unfolded into one 
plane would appear as in Fig. 72. Other points on the perpendicular 
projection of the intersection would be found in the same manner. 

Points for a curve made by cutting a cone and a plane are located 
by the projection of intersecting elements. 



Figures 73, 74, 75, are lettered alike and the explanation will apply 
to each one. The vertical plane projection of cone and cutting plane are 
given. Project from this, horizontal and perpendicular views of the 
cone. Select some element of the cone as A" B" and project it to the 
perpendicular view by way of the element in the horizontal. The height 
of the intersection G" projected directly to the P. P. and gives G'" a point 
on the P. P. curve Another way slightly different is to select on the H.P. 
some element of the intersecting plane as D' E' and an element of the 
cone A' F' intersecting it at G', project the plane-element to D'" E'" and 
the cone element to the A" F". Here it intersects the plane element at 


35 
































\ 


G" which projected directly to the P.P. gives G'", a point on the P.P. 
curve. The H. P. curve would evidently pass through the point G'. To 
get points on the curve made on the intersecting plane, use an auxiliary 
plane, parallel to it. To find any point as G'", project G", from the V. 
P. and determine its distance from the center line measured either on 
the P. P. or the H. P. 

Where the plane cuts the cone’s axis of revolution at the same angle 
as does an element of the cone, the intersection forms a parabola. This 
is easily remembered by associating the first syllables of parabola and 
parallel. 

If the plane makes a less angle with the axis of revolution than does 
in element, the curve is an hyperbola. 

In the case of the plane making a greater angle with the axis of revo¬ 
lution than does the element or what is the same thing cuts elements on 
both sides of the axis the curve is an ellipse. 

The intersection of solids which could each be formed by the revo¬ 
lution of an element about an axis of revolution may be illustrated by 
the intersection of a cone and a cylinder as seen in Fig. 76. Select some 
element of the cone A" B" cutting the cylinder. Project A" B" to the 
H. P. at A' B' and to the P. P. at A'" B'". Project the elements of the 
cylinder C" F'' and D" E" to the H. P. and P. P., and where they inter- 



36 














































sect A' B' and A'" B'" will be points on the curves of intersection. Other 
points may be located in a similar manner. 

The intersection of a plane and an object symmetrical about an axis of 
revolution is given in Fig. 77 where we have the projection of an object 
with a pseudo=sphericaI surface having its axis of revolution perpendicu¬ 
lar to the H. P. and an intersecting plane parallel to the PP. 

Select any circle A" C" cutting the plane and project it to the V. P. 
and P. P. Project a point of intersection D' in the H. P. to the projec¬ 
tion of the circle on the P. P. and the intersection D'" will be a point on 
the curve. Locate enough points to draw curve by arc method. The 
same method could be used in determining the curve found by the inter¬ 
section of a sphere and a plane. 



Surfaces of cones, cylinders, and other solids which could be gener¬ 
ated by a straight line element are capable of development into one 
plane. The development of a cylinder would be a rectangular parallelo 
gram with one dimension equal to the height and the other equal to the 
circumference of the cylinder. The development of a right cone would 
be a sector of a circle with radius equal to the element of the cone and 
length of arc equal to the circumference of the base. The development 
of intersecting solids is illustrated in Fig. 78 by the intersection of a 
cylinder and cone. Draw any element A" B" of the cone intersecting 
the cylinder at C". Project C" to a line parallel to the outline element 
of the cone. Project the element to A' B'. Consider the cone opened 

% 

37 


N 


















































on the element A' D'. With A as a center and radius AD, strike an arc 
D E (for the half development) and step off with bow dividers D E equal 
in length to the arc D' E'. Locate the element AB by making the arc 
D B equal to D' B\ The element A" B" intersecting cylinder at C" is not 
seen in its true length, hence it would not be correct to lay off C B equal 
to C" B",we therefore project it to the element A D where it is shown in its 
true length. C" revolved to the element A B for its intersection gives 
the point C on the developed opening. Selecting other elements on 
the V P and treating them in a similar manner, other points may be 
located on the developed intersection. 



In developing the cylinder it may be separated at some element as 
at F" and a length from F laid off equal to its circumference. The ele¬ 
ment of the cylinder which cuts A" B" at C" is located on the develop¬ 
ment by laying off F C equal to F"C". The intersection of these two 
elements is seen at C' on the H P projected from the V P and the 
point C' projected to the development of the cylinder, gives C a point 
on the developed intersection. In a similar way other points may be 
located and the curve approximated by drawing tangential arcs. The 
development can of course be placed anywhere on the drawing where 
there is room for the entire development. 

In selecting elements it is well to treat one element at a time and be 
guided as to their frequency by the requirements shown after each one is 
finished. 


38 








































Toothed gearing may be considered as having developed from the 
transmission of rotary motion by objects revolving about fixed axes and 
depending on friction for transmission of motion. Take for example two 
cylinders made of wood mounted on axes parallel to each other and at a 
distance apart equal to the sum of the radii of the cylinders. Rotating 
one would cause the other to turn and if they had the same circumfer¬ 
ences they would make complete revolutions in equal periods of time. 
If they had different circumferences they would rotate in periods in¬ 
versely to their diameters as the circumferences are proportional to the 
diameters and also to the radii. If power were to be transmitted the 
surfaces would probably slip on each other and some means would be 
required to prevent it. If pins were driven in at equal distances on each 
cylinder and cavities made between them, we would have crude gearing, 
but if the pins were straight there would be periods when the tip of a 
pin on one cylinder driving on the base of a pin on the other cylinder 
would cause a different ratio of rotation than that based on the diameters 
of the cylinders and also varying as the contact approached the line of 
centers and receded from it. Another difficulty arising from this would 
be that not more than one set of teeth would be in contact at one time. 
Certain forms of teeth have been found to give constant rotative speed 
and the curves in common use are known as cycloidal and involute. The 
theory of these curves and many other important considerations in gear¬ 
ing are properly discussed in kinematics and some of the practical ap¬ 
plications only will be taken up here. 



The construction of toothed gearing in the shop may be classed un¬ 
der cut gearing and cast gearing. For cut gearing a blank is prepared 


39 



















































and then cut on a gear cutting machine, milling machine, shaper, or 
planer. Ordinarily it is sufficient for the draftsman to indicate on the 
drawing the diameter of the blank, the number of teeth, and the diame¬ 
tral pitch, showing the blank with a few conventional teeth of about the 
right proportion. Fig. 79 represents such a case. The machine shop 
equipment should be known to the draftsmen and it is well for him to 
keep a tabulation of cutters if the equipment is incomplete and call only 
for such gears as the shop is capable of cutting. If it is necessary to 
have some tooth for which there is not a proper cutter, he should furnish 
a metal templet for the machinist to make a new cutter by or to approxi¬ 
mate to, by using a cutter which will do the work by resetting the blank 
after the first cutting. 

Where cast gearing 
is to be used, either a 
metal templet should 
be made or proper di¬ 
rections given for ap¬ 
proximating with arcs 
of circles, such as 
shown in Fig. 80. 

The pitch circle 
corresponds to the 
rolling cylinder al¬ 
ready mentio ned. 

The names applied to 



the parts of the 


FIG. 80 

tooth 


|£dp. 


are 


easily 


remem¬ 


bered by associating them with dentist, the Latin for “toothist.” Starting 
at the pitch circle if we add to it we have the addendum, and if we de¬ 
duct, we have the dedendum. The height of addendum and dedendum, 
as the terms are used here are equal and are together the working 
depth of the tooth. They are to some extent arbitrary in amount and 
are expressed in some fraction of the circular pitch or of the diametral 
pitch. The Diametral Pitch is a ratio, the number of teeth divided by 
the diameter of the pitch circle. The Circular Pitch is a distance, the 
circumference of the pitch circle divided by the number of teeth which 
gives the distance of a tooth and a space measured on the pitch circle. 
In either case, however, a stated pitch means a definite distance on the 
pitch circle. 

If N represents the number of teeth and D the pitch diameter N -^D 
equals the diametral pitch and /zD-^N the circular pitch. Multiplying 
these expressions together (N-fD) x (/z D-=-N) we get jt as a result, 
hence dividing /z by the diamatral pitch we get the circular pitch and 
dividing rr by the circular pitch we get the diametral pitch. If the 
teeth were perfectly made they could be the same thickness on the pitch 


40 
















circle as the adjoining space, bat it is usual in cast gearing to make them 
something less to provide for inaccuracies, and the difference between 
the distances is called back lash. This distance is sometimes* called 
clearance, but this term will be used here to designate the distance 
the cavity between teeth is carried beyond the working depth. Refer¬ 
ence to Fig. 82 will illustrate the application of the various terms. 

The symbols are: 

D = Pitch diameter. 

AM = Addendum. 

AD —Addendum circle diameter or outside diameter. 

DM = Dedendum. 

DD = Dedendum circle diameter. 

T = Thickness of tooth on pitch circle. 

RD = Root circle diameter. 

DP = Diametral pitch. 

CP = Circular pitch. 

BD=:Base circle diameter. 

C = Clearance. 

BL = Back lash. 

7^ = 3.1416. 

N = Number of teeth. 

The actual ratio of addendum, back lash and clearance to the cir¬ 
cular pitch and the diametral pitch should depend on the character of 
workmanship, method of construction place of use, etc. For cut teeth 
the ratios given by Brown & Sharpe in their gear catalog are as 
follows: 

Dedendum = addendum = 1-f-DP —.3183 CP. 

Back lash = 0. 

Clearance = .05 CP= .15708-^DP. 

Proportions which can easily be remembered. 

A table of dimensions with their proportions published by Brown & 
Sharpe in Practical Treatise in Gearing, will be found on page 50. 

For cast gearing these proportions would answer very well except 
that the back lash should be made sufficient to accomodate the probable 
inaccuracies in moulding. If made 5 % of the Circular Pitch or about 
one eighth on the addendum we will have for the thickness of the tooth, 

T = .48 CP= 1.54-DP. 

The two curves which have been found to best serve the purpose 
for gear teeth are the involute and cycloid. The involute curve may be 
generated by the free end of a string unwinding from a base circle. The 
cycloidal curve is traced by a point on a generating circle rolling inside 
or outside of a pitch circle. If outside it is called epicycloid, and if in- . 
side hypocycloid. Methods of constructing these curves are given on 


41 


pages 28-30. The base circle for the involute curve is found by locating 
the desired line of action or tooth contact, and drawing a circle tangent 
to it. In Fig. 82 the pitch circle, line of action, and base circle are 
shown. The angle between a perpendicular to the line of centers and 
the line of action is usually made 15°. It is sufficiently accurate to lay 
off inside the pitch circle a distance g 1 ^ of the pitch circle radius to draw 
the base circle through. All involute gears of the same pitch laid off 
with the same angle of obliquity are interchangeable and they have also 
the peculiar characteristic of working well together when the centers are 
not exactly the distance of the sum of the radii of the pitch circles. Page 
49 shows a number of involute teeth of actual size marked with diame¬ 
tral and some also with circular pitch. 

In cycloidal gearing a set may be made interchangeable by having 
all the gears of the same pitch and the teeth generated by the same gen¬ 
erating circle. By drawing hypocycloids with varying diameters of 
generating circles which form the the dedendum flank of the tooth, it will 
be observed that when the generating circle is one-half the diameter of 
the pitch circle the curve is a straight line and radial; if greater than one- 
half the pitch circle the curve under cuts, and if less than one-half, the 
tooth is thicker at the base than at the pitch circle. It is usual to make 
the flank of the smallest gear of the set radial, which fixes the generating 
circle at half the pitch diameter of the smallest gear. For the large 
mating gear or rack the addendum or face of the tooth generated by this 
smallest gear generating circle will be narrow at the.top or addendum line. 

If but two gears are to be made it would be better to adopt one gen¬ 
erating circle for the face of one and 
the flank of the other, and another 
generating circle for the other face 
and flank, each being about three- 
eighths the diameter of the pitch 
circles in which they roll. 

The line of action or path of con= 
tact in cycloidal gearing is always 
on a portion of the generating circle 
and is terminated by the addendum 
circle. 

In Figure 81 contact begins at B, 
follows the describing circle to A, 
continues to C and terminates. 

From B to A is the approach, and 
from A to C the recess. The angles 
formed between lines running 
through the c e points to the pitch circle centers and the line of cen¬ 
ters are called angle of approach and angle of recess, and the arcs 



42 





measured by these angles on the pitch circles are called arc of 
approach D A and arc of recess AE, and together arc of contact 
or arc of action D E. If continous rotation is to be had it is 
clearly necessary that this arc of contact shall be as grfat as the 
circular pitch. Figure 81 shows a portion of a pair of 12 teeth 
8 D P gears with generating circles one-third the pitch diameters. 
It is quite evident that very much reduction in the number of 
teeth would increase the circular pitch so much that contact between 
a pair of teeth would cease before another pair began contact. Increas¬ 
ing the size of the generating circles would increase the arc of contact as 
would also lengthening the teeth, thereby increasing the addendum 
circles. 

In Involute gearing the line of action or line of contact is a straight 
line terminating where it comes tangent to the base circles as at B and 
C in Figure 82. The arc of action D A E is measured on the pitch 
circles as in cycloidal gearing and the tooth shown could not have a 

larger circular pitch than that shown and si ill 
have contact between an approaching pair of 
teeth before the preceding ones had parted. 
The example shown is a 12 tooth 3 D P pinion. 
The line of contact could be increased only by 
increasing the angle, which may be done for 
special gearing requiring less than 12 teeth. 
The dedendum outline or flank inside the 
base circle is customarily made radial. As 
the involute curve does not go below the base 
circle, it is not necessary for the mating gear 
to have an addendum greater than one which 
would pass through B in Figure 82, which 
would reduce the dedendum necessary on the 
gear shown and result in stronger teeth. The 
customary proportions are however desirable 
from the considerations of interch nge- 
ability. 

When the radius of the pitch circle in the mating gear becomes very 
large the addendum of its teeth interfere as shown in Figure 83 of a 2 D P 
12 tooth gear and a rack. If the pinion has been made on the cycloidal 
system with radial flanks the correct form of the addendum of the rack 
would be an encycloid formed by a generating circle one-half the 
diameter of the pitch circle of the small gear or pinion. This curve ap¬ 
plied to the rack passing through a point on the edge of the tooth the 
same distance from the pitch line as C, would prevent interference by a 
mixture of the systems. The involute system would be kept intact by 


<</ 



43 










cutting off the tooth at a height determined by the intersection of the line 
of centers and an addendum circle passing through C. 

G. B. Grant in his Treatise on Gear 
Wheels gives for the rack-tooth correc¬ 
tion, a radius of 2.1 inches divided by the 
D P struck from a center on the pitch 
line and through a point one quarter the 
whole depth of the. tooth from the 
addendum. 

To test an involute for interference 
such as occurs in Figure 83. Through 
the point C, swing a circle from the cen¬ 
ter of the gear under consideration and it 
will give an addendum which will not 
interfere. Pass an epicycloid of gener¬ 
ating circle one-half the 12 tooth pinion, 
through the intersection of the noninter¬ 
fering addendum circle and the involute, 
and if it cuts into the involute interfer¬ 
ence will occur, but if it passes outside 
of it there will be no interference. 

When bevel gearing is required 
the i roblem is to construct teeth on 
two pitch cones whose bases are proportioned properly to give the 
r otative ratio and whose heights are such that the axes form with each 

other the angle de¬ 
sired between the 
lines of shafting. 
Figure 84 shows two 
pitch cones with their 
apeces meeting at the 
center of a sphere and 
a generating cone 
rolling on oneof them. 
If the generating 
r, *c cone had rolled from 
a contact at A the 
element o f contact 
would describe an 
epicycloid. If it had 
rolled inside the pitch 
cone it w r ould have 
described a hypo- 
cycloid, and these surfaces formed into teeth would give correct 




44 

































































gearing. It is quite evident that the element A O having one end at the 
center of the sphere would have its other end moving in the surface of 
the sphere. If it were possible to develop the surface of the sphere into 
a plane, construct the curve A C as in spur gearing and put it back on 
the sphere we would have the line to follow with a tool always travelling 

toward O. The sphere not being a de¬ 
velopable surface we resort to an approx¬ 
imation known as Tredgold’s method. 
In Figure 85 the cones A O B and BOC 
are tangent along the line O B and have 
their bases AB and B C in the surface of 
a sphere. Two tangent cones are drawn 
so that AD makes 90° with AO and 
EC 90° with CO. These cones can 
be developed and the circle formed by their basis used as pitch circles, 
the ordinary involute teeth laid out on them and the cones rolled up 
placed back on the pitch cones the outline of the teeth pricked through 
on the blanks ready for machining. 




Figure 86 shows the me hod of actually drawing the templet. A OB, 
and BOC are two pitch cones which have bases AB and BC propor 


45 





































tional to the desired speed ratio. The tangent cones ADB and BEC 

are drawn and developed. The circum¬ 
ference corresponding to A B is divided 
by the number of teeth and the circular 
pitch so formed stepped off on the pitch 
circle of the developed cone and the forms of 
teeth drawn as in spur gearing. The dimen¬ 
sions needed for preparing the blank are 
shown in Figure 87, and can be scaled from 
the construction and put on. Where the 
bevel gear teeth are to be milled with stock 
cutters the same kind of a blank is used. 
It is quite evident that a milled tooth will 
give a very imperfect approximation to the 
correct shape. 



For laying out angles or for measuring 
angles the table of sines is given. The 
decimal fraction in the table is the value of 
B D A B in Figure 88. To lay off any 
angle take the sign of half the desired angle, A 
multiply it by twice the radius AB and on 
the arc struck with A B, lay off the amount. 

By using a radius of 5" the calculation is 
very simple. 

FIG. 88 

Example: Required an angle of 15 . From the table, the sign of 
half 15 that is 7 : -30' is. 1305. This multiplied by 2 X 5 (taking a radius 
of 5") gives 1 305. Strike an arc with 5" radius and with B as a center 
strike another arc of 1.305" radius intersecting as at C, then BAC 
will be 15°. 

To measure any given angle as BAC. Strike arc with radius A B 
and divide B C by twice the radius A B, the result will be the sign of 
hdf the angle. Double this angle and the result will be the measure of 
the angle BAC. 

Example: Required the measure of the angle BAC. Strike an arc 
B C with A B equal to 5", measure B C which will be assumed to be 3.9", 
this divided by 2 X 5 will be .39. Looking in the table this corresponds 
to 23 , which doubled equals 46° the measure of BAC. 



46 






















TABLE OF NATURAL SINES FOR EVERY 10' TO 90° 


Sine. 

)eg 

Min. 

Sine. 

Deg 

din. 

Sine. 

Deg 

Min, 

Sine. 

Deg. 

Min, 

Sine. 

Deg 

Min. 

Sine. 

Deg 

Min 

Sine. 

Deg 

Min 

Sine. 

.1736 

1765 

1794 

20 

00 

10 

20 

.3420 

.3448 

.3475 

30 

00 

10 

20 

. 5000 
.5025 
.5050 

40 

00 

10 

20 

.6428 

.6450 

.6472 

50 

»'J 

10 

20 

.7660 

. 7679 
7698 

60 

00 

10 

20 

.8660 

.8675 

.8689 

70 

00 

10 

20 

.9397 

.9407 

.9417 

BO 

00 

10 

20 

.8848 

.9853 

.9858 

1822 
. 1851 
.1880 


30 

40 

50 

.3502 

.3529 

.3557 


30 

40 

50 

.5075 

.5100 

.5125 


30 

40 

50 

.6494 

.6517 

.6539 


30 

40 

50 

7716 
, 7735 
.7753 


30 

40 

50 

.8704 

.8718 

.8732 


30 

40 

50 

. 9426 
.9430 
.9446 


30 

40 

50 

.9863 

.9868 

.9872 

1908 
.1937 
. 1965 

21 

00 

10 

20 

.3584 

.3611 

.3638 

31 

00 

10 

20 

.5150 

.5175 

.5200 

41 

00 

10 

20 

.6561 

.6583 

.6604 

51 

00 

10 

20 

. 7771 
.7790 
: 7808 

61 

00 

10 

20 

.8746 

.8760 

.8774 

71 

00 

10 

20 

.9455 

.9465 

.9474 

81 

00 

10 

20 

.9877 

.9881 

.9886 

1994 

2022 

.2051 


30 

40 

50 

.3665 

.3692 

.3719 


30 

40 

50 

. 5225 
.5250 
.5275 


30 

40 

50 

.6626 

.6648 

.6670 


30 

40 

50 

7826 

7*44 

.7802 


30 

40 

50 

.8788 

.8802 

.8816 


30 

40 

50 

9483 

.9492 

9502 


30 

40 

50 

.9890 

.9894 

.9899 

.2079 

.2108 

.'2136 

22 

00 

10 

20 

.3746 

.3773 

.3800 

32 

00 

10 

20 

. 529!) 
.5324 
.5348 

42 

00 

10 

20 

.6691 

.6713 

.6734 

52 

00 

10 

20 

.7880 

.7898 

.7916 

62 

00 

10 

20 

.8829 

.8813 

.8857 

72 

00 

10 

20 

.9511- 

.9520 

9528 

82 

00 

10 

20 

.9903 

.9907 

.9911 

.2164 

.2193 

.2221 


30 

40 

50 

.3827 

.3854 

.3881 


30 

40 

50 

.5373 

.5398 

.5422 


30 

40 

50 

.6756 

.6777 

.6799 


30 

40 

50 

. 7934 
.7951 
.7969 


30 

40 

50 

.8870 

.8884 

.8897 


30 

40 

50 

.9537 
.9546 
. 9555 


30 

40 

50 

.9914 

.9918 

.9922 

.2250 
. 2278 
.2306 

23 

00 

10 

2') 

.3907 

.3934 

.3961 

33 

00 

10 

20 

.5446 

.5471 

.5495 

43 

00 

10 

20 

.6820 

.6841 

.6862 

53 

00 

10 . 

20 

.7986 
. 8004 
.8021 

63 

00 

10 

20 

.8910 

.8923 

.8936 

73 

00 

10 

20 

.9563 

9572 

9580 

83 

00 

10 

20 

.9925 

.9929 

.9932 

2334 

.2363 

.2391 


30 

40 

50 

.3987 

.4014 

.4041 


30 

40 

50 

.5519 

.5544 

.5568 


30 

40 

50 

.6884 

.6905 

.6926 


30 

40 

50 

.8039 

.8056 

.8073 


30 

40 

50 

.8949 

.8962 

.8975 


30 

40 

50 

9588 
. 9596 
.9605 


30 

40 

50 

.9936 

.9939 

.9942 

.2419 

2447 

.2476 

24 

00 

10 

20 

.4067 

.4094 

.4120 

34 

00 

10 

20 

. 5592 
.5616 
.5640 

44 

00 

10 

20 

.6947 

.6967 

.6988 

54 

00 

10 

20 

8090 

.8107 

8124 

64 

00 

10 

20 

.8988 

.9001 

.9013 

74 

00 

10 

20 

.9613 

.9621 

.9628 

84 

00 

10 

20 

.9945 

.9948 

.9951 

.2504 
. 2532 
.2560 

.258* 

.2616 

.2644 

25 

30 

40 

50 

00 

10 

20 

.4147 

.4173 

.4200 

.4226 

.4253 

.4279 

35 

30 

40 

50 

00 

10 

20 

.5664 
. 568* 
5712 
.5736 
.5760 
.5783 

45 

30 

40 

5Q 

00 

10 

20 

.7009 

.7030 

.7050 

.7071 

.7092 

.7112 

55 

30 

40 

50 

00 

10 

20 

.8141 

.8158 

.8175 

.8192 

8208 

.8225 

65 

30 

40 

50 

00 

10 

20 

.9026 

.9038 

.9051 

.9063 

.9075 

.9088 

75 

30 

40 

50 

00 

10 

20 

9636 
.9644 
9852 
9659 
. 9667 
9674 

85 

30 

40 

50 

00 

10 

20 

.9954 

.9957 

.9959 

.9962 

.9964 

.9967 

.2672 

.2700 

.2728 


30 

40 

50 

.4305 

.4331 

.4358 


30 

40 

50 

.5807 

.5831 

.5854 


30 

40 

50 

7133 

.7153 

.7178 


30 

40 

50 

.8241 

8258 

8274 


30 

40 

50 

.9100 

.9112 

.9124 


30 

40 

50 

9681 

96*9 

9696 


30 

40 

50 

.9969 

.9971 

.9974 

.2756 

.2784 

.2812 

26 

00 

10 

20 

.4384 

.4410 

.4436 

36 

00 

10 

20 

.5878 

.5901 

.5925 

46 

00 

10 

20 

.7193 

.7214 

.7234 

56 

00 

10 

20 

8290 

*307 

8323 

66 

00 

10 

20 

.9135 

.9147 

.9159 

76 

00 

10 

20 

9703 

.9710 

.9717 

80 

00 

10 

20 

.9976 

.9978 

.9980 

.2840 
. 2868 
.2896 


30 

40 

50 

.4462 

.4488 

.4514 


30 

40 

50 

.5948 

5972 

.5995 


30 

40 

50 

.7254 

.7274 

.7294 


30 

40 

50 

.8339 

.8355 

.8371 


30 

40 

50 

.9171 

.9182 

.9194 


30 

40 

50 

9724 
. 9730 
9737 


30 

40 

50 

.9981. 

.9983 

.9985 

.2924 

.2952 

.2979 

27 

00 

10 

20 

.4540 

.4566 

.4592 

37 

00 

10 

20 

6018 

.6041 

.6065 

47 

00 

10 

20 

.7314 

.7333 

.7353 

57 

00 

10 

20 

.8387 

.8403 

.8418 

67 

00 

10 

20 

.9205 

.9216 

.9228 

77 

00 

10 

20 

.9744 
9750 
. 9757 

87 

00 

10 

20 

.9986 

.9988 

.9989 

.3007 

.3035 

.3062 


30 

40 

50 

.4617 

.4643 

.4669 


30 

40 

50 

.6088 

.6111 

.613-1 


30 

40 

.50 

.7373 

.7392 

.7412 


30 

40 

50 

.8434 

.8450 

.8465 


30 

40 

50 

.9239 

.9250 

.9261 


30 

40 

50 

9763 
9769 
. 9775 


30 

40 

50 

9990 

.9992 

.9993 

.3090 

.3118 

.3145 

28 

00 

10 

20 

% 4695 
4720 
.4746 

38 

00 

10 

20 

.6157 

6180 

6202 

48 

00 

10 

20 

.7431 

.7451 

.7470 

58 

00 

10 

20 

8480 

8496 

.8511 

68 

00 

10 

20 

-.9272 

.9283 

.9293 

78 

00 

10 

20 

9781 

.9787 

9793 

88 

00 

10 

20 

.9994 

.9995 

.9996 

.3173 

.3201 

.3228 


30 

40 

50 

.4772 

.4797 

.4823 


30 

40 

50 

6225 

6248 

.6271 


30 

40 

50 

.7490 

.7509 

.7528 


30 

40 

50 

.8526 

.8542 

.8557 


30 

40 

50 

.9304 

.9315 

.9325 


30 

40 

50 

9799 
. 9805 
9811 


30 

40 

50 

.9997 

.9997 

.9998 

. 3256 
.32*3 
3311 

29 

00 

10 

20 

.4848 

.4874 

.4899 

39 

00 

10 

20 

.6293 

.6316 

.6338 

49 

00 

10 

20 

.7547 

.7566 

.7585 

59 

00 

10 

20 

.8572 

.8587 

.8601 

69 

00 

10 

20 

.9336 

.9346 

.9356 

70 

00 

10 

20 

.9816 

9*22 

.9827 

8£ 

00 

10 

20 

.9998 

.9999 

.9999 

.3338 

3365 

.3393 


30 

40 

50 

.4924 

.4950 

.4975 


30 

40 

50 

6361 

63*3 

.6400 


80 

40 

50 

.7604 

.7623 

.7642 


30 

40 

.50 

8616 

.8631 
. 8640 


30 

40 

50 

.9367 

.9377 

.9387 


30 

40 

50 

.9833 

.9838 

.9843 


30 

40 

50 

1.0000 

1.0000 

1.0000 


Deg. 


Min 


Sine. 


Deg. Min, 


00 

10 

20 

30 

40 

50 

00 

10 

20 

30 

40 

50 

00 

10 

20 

30 

40 

50 

00 

10 

20 

30 

40 

50 

00 

10 

20 

30 

40 

50 

00 

10 

20 

80 

40 

50 

00 

10 

20 

30 

4o 

50 

00 

10 

20 

30 

40 

50 

00 

10 

20 

30 

40 

50 

00 

10 

20 

30 

40 

50 


.0000 

.0029 

.0058 

.0087 

.0116 

.0145 

.0175 

.0204 

.0233 

.0262 

.0291 

.0320 

.0349 

.0378 

.0407 

.0436 

.0465 

.0494 

.0523 

.0552 

.0581 

.0610 

.0640 

.0669 

.0698 

.0727 

.0756 

.0785 

.0814 

.0843 

.0872 

.0901 

.0929 

.0958 

.09* 

.1016 

.1045 

.1074 

.1103 

.1132 

.1161 

.1190 

.1219 

.124* 

.1276 

.1305 

.1334 

.1363 

.1392 

.1421 

.1449 

.1478 

.1507 

.1536 

.1564 

.1593 

.1622 

.1650 

.1679 

.1708 


10 


11 


12 


13 


14 


15 


16 


18 


19 


00 

10 

•in 

30 

40 

50 

00 

10 

20 

30 

40 

50 

00 

10 

20 

30 

40 

50 

00 

10 

20 

30 

40 

50 

00 

10 

20 

30 

10 

50 

00 

10 

20 

30 

40 

50 

00 

10 

20 

30 

40 

50 

00 

10 

20 

30 

40 

50 

00 

10 

20 

30 

40 

50 

00 

10 

20 

30 

40 

50 


47 






























































A method of approximation by arcs for forming gear teeth has 
been devised by Mr. George B. Grant. The system for the involute 
tooth is tabulated below. The curves were obtained by making correct 
involute teeth 8" long and finding, by trial, centers on the base line 
which give a very close approximation. Interference was provided 
against by making the point of teeth epicycloidal to work with radial 
flank of 12 tooth pinion. The two radii used are as indicated in Figure 
• 80 . 

grant’s involute odontograph. 


Taken, by Permission, from Grant's Treatise on Gear Wheels. 
Standard Interchangeable Tooth, Centers on Base Line 


Teeth. 

Divide by the 
Diametral Pitch. 

Multiply by the 
Circular Pitch. 

Face 

Radius. 

Flank 

Radius. 

Face 

Radius. 

Flank 

Radius 

10 

2.28 

.69 

n .73 

.22 

11 

2.40 

.83 

.76 

.27 

12 

2.51 

.96 

.80 

.31 

13 

2.62 

1.09 

.83 

.34 

14 

2.72 

1.22 

.87 

.39 

15 

2.82 

1.34 

.90 

.43 

16 

2.92 

1.46 

.93 

.47 

17 

3.02 

1.58 

. 96 

.50 

18 

3.12 

1.69 

.99 

.54 

19 

3.22 

1.79 

1.03 

.57 

20 

3.32 

1.89 

1.06 

.60 

21 

3.41 

1.98 

1.09 

.63 

22 

3.49 

2.06 

1.11 

.66 

23 

3.57 

2.15 

1.13 

.69 

24 

3.64 

2.24 

1.16 

.71 

25 

3.71 

2.33 

1.18 

.74 

26 

3.7S 

2 42 

1.20 

. 77 

27 

3.85 

2 50 

1.23 

.80 

28 

3.92 

2.59 

1.25 

.82 

29 

3.99 

2 67 

1.27 

.85 

30 

4.06 

2.76 

1.29 

.88 

31 

4.13 

2.85 

1.31 

.91 

32 

4.20 

2.93 

1.34 

.93 

33 

4.27 

3.01 

1.36 

.96 

34 

4.33 

3.09 

1.38 

.99 

35 

4.39 

3.16 

1.39 

1.01 

36 

4.45 

3.23 

1.41 

1.03 

37—40 

4. 

20 

1 . 

34 

4 l—45 

4.63 

1 . 

48 

46 — 51 

5 06 

1 . 

61 

52 — 60 

5.74 

1 . 

83 

61—70 

6.52 

2. 

07 

71—90 

H 

7 

72 

2. 

46 

91-120 

9. 

78 

3. 

11 

121 — 180 

13.38 

4. 

26 

181-360 

21.62 

6. 

88 


Draw the rack tooth by the special method. 


4 X 


























INVOLUTE TEETH DRAWN TO ACTUAL SIZE 



-\— 3.141B 


r.iOH - 



| DP 



fVA/ly 4 

20 OP 

fjvv 

18 OP 


DP 



16 OP 



2 DP 





14 OP 




8 OP 


4!> 


3 0 P 













































DIMENSIONS OF TEETH ACCORDING TO B. & S. PROPORTIONS 



D P 

C P 

T 

AM 

AM+DM 

DM+C 

AM+D M 
4-C 

i 

G 2832 

3.1416 

2.0000 

4.0000 

2.3142 

4.3142 

a. 

4 

4 1888 

2.0944 

1.3333 

2 6666 

1.5428 

2.8761 

1 

3.141G 

1 5708 

1.0000 

2.0000 

1.1571 

2.1571 

n 

2.5133 

1 2566 

.8000 

1.6000 

.9257 

1.7257 

H 

2 0944 

1.0472 

.6666 

1.3333 

.7714 

1.4381 

If 

1.7952 

.8976 

. 5 714 

i 1429 

.6612 

1.2326 

2 

1 5708 

.7854 

.5000 

1.0000 

. 5785 

1.0785 

2* 

1.3963 

.6981 

.4444 

.8888 

.5143 

.9587 

24 

1.2566 

. 6283 

4000 

.8000 

. 4628 

.8628 

oa 

“4 

1.1424 

5712 

. 3636 

.7273 

.4208 

.7844 

3 

1.0472 

. 5236 

. 3333 

. 6666 

.3857 

.7190 

— 

CO 

. 897G 

.4488 

.2857 

.5714 

3306 

.6163 

4 

. 7854 

.3927 

2500 

. 5000 

.2893 

.5393 

5 

. 6283 

.3142 

.2000 

.4000 

. 2314 

.4314 

G 

. 523G 

.2618 

. 1666 

'.3333 

1928 

.3595 

r* 

i 

.4488 

.2244 

. 1429 

. 2857 

. 1653 

.3081 

8 

.3927 

1963 

.1250 

.2500 

.1446 

.2696 

9 

3191 

1745 

.1111 

.2222 

.1286 

.2397 

10 

.3142 

.1571 

.1000 

.2000 

.1157 

.2157 

11 

. 2856 

.1428 

.0909 

.1818 

.1052 

.1961 

12 

.2618 

.1309 

0833 

.1666 

.0964 

.1798 

13 

.2417 

.1208 

.0769 

.1538 

.0890 

1659 

14 

.2244 

.1122 

.0714 

. 1429 

. 0S26 

.1541 

la 

.2094 

.1047 

.0666 

.1333 

.0771 

.1438 

1G 

.1963 

0982 

.0625 

. 1250 

. 0723 

. 1348 

17 

.1848 

.0924 

.05 8 

.1176 

.0681 

. 1269 

18 

. 1745 

. 0873 

. 0555 

.1111 

.6643 

.1198 

19 

.1653 

.0827 

.0526 

.1053 

.0609 

.1135 

20 

.1571 

.0785 

.0500 

1000 

. 0579 

.1079 

22 

. 1428 

0714 

. 0455 

.0909 

.0526 

.0980 

24 

.1309 

.0654 

.0117 

. 0833 

.0482 

.0898 

2G 

.1208 

.0604 

.0385 

.0769 

.0445 

.0829 

28 

1122 

.0561 

.0357 

.0714 

.0413 

.0770 

30 

. 1047 

.0524 

.0333 

. 066G 

.0386 

.0719 

32 

0982 

0491 . 

.0312 

0625 

.0362 

.0674 


50 






































51 











































































































STANDARD CAP SCREWS 



DIAM. OFHEAD 
HEIGHT OF HEAD 
THREADS PER IN, 


L 


DIAM. OF HEAD 
HEIGHT OF HEAD 
THREADS PER IN 


7-16 
I --4 
QO 


7-IS 
7-32 


3—© 
i —4 


15-32 

SO 


3-8 
I —4 
SO 


3-0 
I --4 


5 

16 


DIAM. OFHEAD 
HEIGHT OF HEAD 
THREADS PER IIM 


I -Q 
3—is 
i© 


9-16 
*- 
II 


“7-IS 

5-16 

IS 


5-Q 

IQ 


“7-IS 
5-16 
I© 


-7-16 

5-16 

1© 


3 

a 


DIAM. OF HEAD 
HEIGHT OF HEAD 
THREADS PER IN 


3- IG 
3-0 
IS 


5-a 
s-is 
16 


3 — IS 
3-S 
16 


3-^4 

IS 


9-16 

3-0 

16 


I -Q 

3-0 

IS 


z 

16 


DIAM. OF HEAD 
HEIGHT OF HEAD 
THREADS PER IN 


3-© 

7—16 

1-4 




1-4 


5-8 

7-IS 

1-4 


13-16 

1-4 


5-0 
*7- IS 
1-4 


Q- IS 
7-16 
1-4 


DIAM. OF HEAD 
HEIGHT OF HEAD 
THREADS PER IN, 


3--4- 
I -2 
IS 


13-16 

13-32 

12 


3-^4 

i-o 

IQ 


7-0 


3 —4 
I -Q 
122 


5-0 
I-Q 
IS 


9 

16 


DIAM. OF HEAD 
HEIGHT OF HEAD 
THREADS PER IN, 


13*16 

0-16 

IQ 


IS- IS 
IS-36 
IQ 


13-16 

9-16 




13-16 

9-16 

IQ 


I I -IS 
6-16 


5 

6 


DIAM. OF HEAD 
HEIGHT OF HEAD 
THREADS PER IN 


7-0 


I I 


1 - 


I I 


7-Q 
S-O 
I I 


I — 1-0 

I I 


7-0 
3-0 
I I 


3 --4 
5-0 
11 


11 

16 


DIAM. OF HEAD 
HEIGHT OF HEAD 
THREADS PER IN 


IS-16 
I I -16 
I I 


I 

I- 1 

I I 


IS-IS 
I I -16 
I I 


I ~ 1—4 

I I 


IS-IG 
I I -16 
I I 


13-16 
I 1-16 
I I 


3 

-4 


DIAM. OF HEAD 
HEIGHT OF HEAD 
THREADS PER IN 


I 

3—4 

IO 


I— 1-4 
0-0 
IO 


I 

3--4 
IO 


I — 


IO 


I 

3—4 

IO 


7-0 

3—4 

IO 


13 

IG 


DIAM. OF HEAD 
HEIGHT OF HEAD 
THREADS PER IN 


I— 1-16 
13-16 
IO 


I- 3-0 

II- 16 
IO 


I— 1-16 
13-10 
IO 


I — I-; 

IO 


I— 1-16 
1316 
IO 


I—1-16 
13-16 
IO 


z 

a 


DIAM. OF HEAD 
HEIGHT OF HEAD 
THREADS PER IN 


I— 1-0 

7-0 

O 


I— l-Q 
3—4 


I — 1-8 
7-8 


i— i-; 


I — I-16 
13-16 


I— 1-16 

*3-16 

O 


15 

16 


DIAM. OF HEAD 
HEIGHT OF HEAD 
THREADS PER IN 


I — 3-16 
*S - 16 
9 


I — 9-16 
25-32 


1-3-16 

15-16 

& 


I—• 1-16 


I —3-16 
15-16 


1-3-16 
15-16 


DIAM. OF HEAD 
HEIGHT OF HEAD 
THREADS PER IN 


I— 1—4 


1-3-8 

13-16 

O 


I — 1—4 
I 

© 


I-13-16 

O 


I — 1-^4 
I 

© 


I—I--4 

I 

& 


54 




















































































Decimal 

Equivalents 


Standard Pipe Threads. 


<-L 



INSIDE 

DIAM 

NOMINAL 

INSIDE 

DIAM. 

ACTUAL 

D 

OUTSIDE 

DIAM. 

ACTUAL 

D' 

DIAMETERS 0F TH’d. 
ATEND~PIPE 

LEN6THS<*THREAD 

THREADS 

PERIN. 

DRILLS 

80RE 

FULL 

L' 

FULL AT 

ROOT 

[L 

TOTAL 

L 

ROOT 

D" 

CROWN 

D" 

1-8 

.27 

.405 

342 

.393 

.19 

.2 6 

.30 

27 

1 1-32 

1 -A 

.354 

54 

.442 

.522 

.29 

.4 

.62 

18 

29-64 

3-8 

.49 

.67 

.5 8 

.656 

.30 

.4 1 

.63 

16 

1 9-32 

1-2 

.625 

84 

.7 i 7 

.8 l 6 

.39 

.53 

.6 2 

K 

2 3-32 

3'4 

.82 

105 

.926 

1 .025 

40 

.54 

.8 3 

14 

15-16 

1 

1.04 

i.3i 

1 .1 6' 

1.28 

.51 

.68 

1.02 

l 1,1-2 

1 , 3-16 

1. 1-4 

1.38 

1.66 

1 .506 

1 .626 

.54 

.7 1 

I.OS 

11.1-2 

l, 1-2 

1. 1-2 

l.8l 

1.90 

l 7-46 

1.866 

.55 

.7 2 

1.06 

11,1-2 

l, 3-4 

2 

2.067 

2.375 

2.21 9 

2.338 

.58 

.7 5 

l.l 0 

II, 1-2 

2. 7-32 

2.1-2 

2.468 

2 875 

2.647 

2.8i 9 

.89 

1.14 

1-64 

8 

2.21-32 

3 

3.067 

3.5 

3.268 

344 

.95 

1.2 

1.70 

8 

3. 9-32 

3. i-2 

3.540 

4.0 

3.765 

3.937 

1.0 

1.25 

1.75 

8 

3.25-32 


4.026 

4.5 

4.265 

4.434 

1.05 

1.3 

1.80 

8 

A , 9-32 

4. 1-2 

4.508 

5 

A .759 

4.931 

l.l 

1.35 

185 

8 

4, 25-32 

5 

5 045 

5.563 

5.32 

5.49 

1.16 

1.41 

1.91 

8 

5. 11-32 

6 

6-065 

6.625 

6.372 

6.547 

1.26 

1.51 

ZO 1 

8 

6, 3-8 

7 

7.023 

7625 

7.37 

7.54 

1.36 

1.61 

211 

8 

7. 3-8 

i 

8.082 

8-625 

8.361 

8.534 

1.46 

1.71 

2.21 

8 

8. 3-8 

9 

9. 

3.688 

9.4 17 

9.59 

1.57 

1.82 

2.32 

8 

9, 7-16 

10 

10-019 

10.75 

10.472 

10.645 

1.68 

1.93 

2.43 

8 

10. 1-2 


Wrot iron pipe is threaded according to 
the Briggs standard which is discussed in the 
Transactions of the American Society of 
Mechanical Engineers, Volume VII, page 
313. The outline of the thread is similar to 


the U. S Standard, excepting that the flat is 
one-tenth the pitch. The length of the full 
thread is= (4.8-f-.8 outside diameter) pitch. 
This is followed by two threads full at root 
and then four imperfect threads. 


32 ds ‘ 

64. ths * 

Decimal. 

Fraction. 

4 

1 

.015625 


1 

2 

.03125 



3 

.046875 


2 

4 

.0625 

1-16 


5 

.078125 


3 

6 

09375 



7 

.109375 


4 

8 

.125 

1-8 


9 

.140625 


5 

10 

.15825 



11 

.171876 


6 

12 

.1875 

3-16 


13 

.203125 


7 

14 

.21875 



15 

.234376 


8 

16 

.25 

1-4 


17 

.265625 


9 

18 

.28125 



19 

.296875 


10 

20 

.3125 

5-16 


21 

.328125 


11 

22 

.34375 



23 

359375 


12 

24 

.375 

3-8 


25 

.390625 


13 

26 

.40625 



27 

.421875 


14 

28 

.4375 

7-16 


29 

.453125 


15 

30 

.46875 



31 

.484375 


16 

32 

.5 

1-2 


33 

.515625 


17 

34 

.53125 



35 

.546875 


18 

36 

.5625 

9-16 


37 

.578125 


19 

38 

.59375 



39 

.009375 


20 

40 

.625 

5-8 


41 

.640625 


21 

42 

.65625 



43 

.671875 


22 

44 

.6875 

11-16 


45 

.703125 


23 

46 

.71875 



47 

.734375 


24 

48 

.75 

3-4 


49 

.765625 


25 

50 

.78125 



61 

.796875 


26 

52 

.8125 

13-16 


53 

.828125 


27 

54 

.84375 



65 

.859375 


28 

56 

•C75 

7-8 


57 

.890625 


29 

58 

.90625 

* 


59 

.921876 


30 

60 

.9375 

16-16 


61 

.953125 


31 

62 

.96876 



63 

.984375 


32 

64 

1 . 

1 


53 











































































to oo os tn ba bo ►-» o io bo is* i- ho bo I— ca io bo ~<J b>cn it»- co bo !-•. o ho bo ^ bn cj< ho bo b- o> iooo^bibt ho bo !—. io coooVjosctt It* ho bo 


CIRCLES 


By tenths to sixteen. 


Circum . 

Area . 

Diam . 

Circum . 

Area . 

Diam . 

Circum . 

Area . 



6.6 

18.8496 

28.2743 

12.0 

37.6991 

113.0973 

.31416 

.007854 

.1 

19.1637 

29.2247 

.1 

38.0133 

114.9901 

.62832 

.031416 

.2 

19.4779 

30.1907 

.2 

38.3274 

116.8987 

.94248 

.070686 

.3 

19.7920 

• 31.1725 

.3 

38.6416 

118.8229 

1.2566 

.12566 

.4 

20.1062 

32.1699 

.4 

38.9557 

120.7628 

1.5708 

.19635 

.5 

20.4204 

33.1831 

.5 

39.2699 

122.7185 

1.8850 

.28274 

.6 

20.7345 

34.2119 

.6 

39.5841 

124.6898 

2.1991 

.38485 

.7 

21.0487 

35.2565 

.7 

39.8982 

126.6769 

2.5133 

.50266 

.8 

21.3628 

36.3168 

.8 

40.2124 

128.6796 

2.8274 

.63617 

.9 

21.6770 

37.3928 

.9 

40.5265 

130.6981 

3.1416 

.7854 

7.0 

21.9911 

38.4845 

13.0 

40.8407 

132.7323 

3.4558 

.9503 

.1 

22.3053 

39.5919 

.1 

41.1549 

134.7822 

3.7699 

1.1310 

.2 

22.6195 

40 . 71.50 

.2 

41.4690 

136.8478 

4.0841 

1.3273 

.3 

22.9336 

41.8539 

.3 

41.7832 

138.9291 

4.3982 

1.5394 

.4 

23.2478 

43.0084 

.4 

42.0973 

141.0261 

4.7124 

1.7671 

.5 

23.5619 

44.1786 

.5 

42.4115 

143.1388 

5.0265 

2.0106 

.6 

23.8761 

45.3646 

.6 

42.7257 

145.2672 

5.3407 

2.2698 

.7 

24.1903 

46.5663 

7 

43.0398 

147.4114 

5.6549 

2.5447 

.8 

24.5044 

47.7836 

.8 

43.3540 

149.5712 

5.9690 

2.8353 

.9 

24.8186 

49.0167 

9 

43.6681 

151.7468 

6.2832 

3.1416 

8.0 

25.1327 

50.2655 

14.0 r 

43.9823 

153.9380 

6.5973 

3.4636 

.1 

25.4469 

51.5300 

.1 

44.2965 

156.1450 

6.9115 

3.8013 

.2 

25.7611 

52.8102 

.2 

44.6106 

158.3677 

7.2257 

4.1548 

.3 

26.0752 

54.1061 

.3 

44.9248 

160.6061 

7.5398 

4.5239 

.4 

26.3894 

55.4177 

.4 

45.2389 

162.8602 

7.8540 

4.9087 

.5 

26.7035 

56.7450 

.5 

45.5531 

165.1300 

8.1681 

5.3093 

.6 

27.0177 

58.0880 

.6 

45.8673 

167.4155 

8.4823 

5.7256 

.7 

27.3319 

59.4468 

.7 

46.1814 

169.7167 

8.7965 

6.1575 

.8 

27.6460 

60.8212 

.8 

46.4956 

172.0336 

9.1106 

6.6052 

.9 

27.9602 

62.2114 

.9 

46.8097 

174.3662 

9.4248 

7.0686 

9.0 

28.2743 

63.6173 

15.0 

47.1239 

176.7146 

9.7389 

7.5477 

.1 

28.5885 

65.0388 

.1 

47.4380 

179 . C 786 

10.0531 

8.0425 

.2 

28.9027 

66.4761 

.2 

47.7522 

181.4584 

10.3673 

8.5530 

.3 

29.2168 

67.9291 

.3 

48.0664 

183.8539 

10.6814 

9.0792 

.4 

29.5310 

69.3978 

.4 

48.3805 

186.2650 

10.9956 

9.6211 

.5 

29.8451 

70.8822 

.5 

48.6947 

188.6919 

11.3097 

10.1788 

.6 

30.1593 

72.3823 

..6 

49.0088 

191.1345 

11.6239 

10.7521 . 

.7 

30.4734 

73.8981 

.7 

49.3230 

193.5928 

11.9381 

11.3411 

.8 

30.7876 

75.4296 

.8 

49.6372 

196.0668 

12.2522 

11.9459 

.9 

31.1018 

76.9769 

.9 

49.9513 

198.5565 

12.5664 

12.5664 

10.0 

31.4159 

78.5398 

98.0 

307.8761 

7542.9640 

12.8805 

13.2025 

-.1 

31.7301 

80.1185 

.1 

308.19021 

7558.3656 

13.1947 

13.8544 

.2 

32.0442 

81.7128 

.2 

308.5044 

7573.7830 

13.5088 

14.5220 

.3 

32.3584 

83.3229 

.3 

308.8186 

7589.2161 

13.8230 

15.2053 

.4 

32.6726 

84.9487 

.4 

309.1327 

7604.6648 

14.1372 

15.9043 

.5 

32.9867 

86.5901 

.5 

309.4469 

7620.1293 

14.4513 

16.6190 

.6 

33.3009 

88.2473 

.6 

309.7610 

7635.6095 

14.7655 

17.3494 

.7 

33.6150 

89.9202 

.7 

310.0752 

7651.1054 

15.0796 

18.0956 

.8 

33.9292 

91.6088 

:8 

310.3894 

7666.6170 

15.3938 

18.8574 

.9 

34.2434 

93.3132 

.9 

310.7035 

7682.1444 

15.7080 

19.6350 

11.0 

34.5575 

95.0332 

99.0 

311.0177 

7697.6893 

16.0221 

20.4282 

.1 

34.8717 

96.7689 

.1 

311.3318 

7713.2461 

16.3363 

21.2372 

.2 

35.1858 

98.5203 

.2 

311.6460 

7728.8206 

16.6504 

22.0618 

.3 

35.5000 

100.2875 

.3 

311.9602 

7744.4107 

16.9646 

22.9022 

.4 

35.8142 

102.0703 

.4 

312.2743 

60.0166 

17.2788 

23.7583 

.5 

36.1283 

103.8689 

5 

312.5885 

7775.6382 

17.5929 

24.6301 

.6 

36.4425 

105.6832 

.6 

312.9026 

7791.2754 

17.9071 

25.5176 

.7 

36.7566 

107.5132 

7 

313.2168 

7806.9284 

18.2212 

26.4208 

.8 

37.0708 

109.3588 

.8 

313.5309 

7822.5971 

18.5354 

27.3397 

.9 

37.3850 

111.2202 

.9 

313.8451 

7838.2815 


54 












CIRCLES 

By thirty-seconds to one, sixteenths to six, eighths to 10. 


Diam . 

Ins. 

ClR- 

CUMF. 

Ins. 

Area. 
Sq. Ins. 

D IA 31. 

Ins. 

ClR- 

CU3IF. 

Ins. 

Area. 
Sq. Ins. 

Diam. 

Ins. 

ClR- 

CUMF 

Ins. 

Area. 
Sq. Ins. 

1 64 

.049087 

.00019 

1 15-16 

6.08684 

2.9483 

4 15-16 

15.5116 

19.147 

1-32 

.098175 

.00077 

2 . 

6.28319 

3.1416 

5 . 

15.7080 

19.635 

3-64 

.147262 

.00173 

1-16 

6.47953 

3.3410 

1-16 

15.9043 

20.129 

1-16 

.196350 

.00307 

1-8 

6.67588 

3.5466 

1-8 

16.1007 

20.629 

3-32 

.294524 

.00690 

3-16 

6.87223 

3.7583 

3-16 

16.2970 

21.135 

1-8 

.392699 

.01227 

1-4 

7.06858 

3.9761 

1-4 

16.4934 

21.648 

5-32 

.490874 

.01917 

5 16 

7.26493 

4.2000 

5-16 

16.6897 

22.166 

3-16 

. 589)49 

.02761 

3-8 

7.46128 

4.4301 

3-8 

16.8861 

22.691 

7-32 

.687223 

.03758 

7-16 

7.65763 

4.6664 

7-16 

17.0824 

23.221 

1-4 

.785398 

.04909 

1-2 

7.85398 

4.9087 

1-2 

17.2788 

23.758 

9-32 

.883573 

.06213 

9-16 

8.05033 

5.1572 

9-16 

17.4751 

24.301 

5-16 

.981748 

.07670 

5-8 

8.24668 

5.4119 

5-8 

17.6715 

24.850 

11-32 

1.07992 

.09281 

11-16 

8.44303 

5.6727 

11-16 

17.8678 

25.406 

3-8 

1.17810 

.11045 

3-4 

8.63938 

5.9396 

3-4 

18.0642 

25.967 

13-32 

1.27627 

.12962 

13-16 

8.83573 

6.2126 

13-16 

18.2605 

26.535 

7-16 

1.37445 

.15033 

7-8 

9 . 032 . '8 

6.4918 

7-8 

18.4569 

27.109 

15-32 

1.47262 

.17257 

15-16 

9.22843 

6.7771 

15-16 

18.6532 

27.688 

1-2 

1.57080 

.19635 

3. 

9.42478 

7.0686 

6. 

18.8496 

28.274 

17-32 

1.66897 

.22166 

1-16 

9.62113 

7.3662 

1-8 

19.2423 

29.465 

9-16 

1.76715 

.24850 

1-8 

9.81748 

7.6699 

1-4 

19.6350 

30.680 

19-32 

1.86532 

.27688 

3-16 

10.0138 

7.9798 

3-8 

20.0277 

31.919 

5-8 

1.96350 

.30680 

1-4 

10.2102 

8.2958 

1-2 

20.4204 

33.183 

21-32 

2.06167 

.33824 

5-16 

10.4065 

8.6179 

5-8 

20.8131 

34.472 

11-16 

2.15984 

.37122 

3-8 

10.6029 

8.9462 

3-4 

21.2058 

35.785 

23-32 

2.25802 

.40574 

7-16 

10.7992 

9.2806 

7-8 

21.5984 

37.122 

3-4 

2.35619 

.44179 

1-2 

10.9956 

9.6211 

7 

21.9911 

38.485 

25-32 

2.45437 

.47937 

9-16 

11.1919 

9.9678 

1-8 

22.3838 

39.871 

13-16 

2.55254 

.51849 

5-8 

11.3883 

10.321 

1-4 

22.7765 

41.282 

27-32 

2.65072 

.55914 

11-16 

11 5846 

10.680 

3-8 

23.1692 

42.718 

7-8 

2.74889 

. 6(1132 

3-4 

11.7810 

11.045 

1-2 

23.5619 

44.179 

29-32 

2.84707 

.64504 

13-16 

11.9773 

11.416 

5-8 

23.9546 

45.664 

15-16 

2.94524 

.69029 

7-8 

12.1737 

11.793 

3-4 

24.3473 

47.173 

31-32 

3.04342 

.73708 

15-16 

12.3700 

12.177 

7-8 

24.7400 

48.707 

1 . 

3.14159 

.78540 

4. 

12.5664 

12.566 

8 . 

25.1327 

50.265 

1-16 

3.33794 

.88664 

1-16 

12.7627 

12.962 

1-8 

25.5254 

51.849 

1-8 

3.53429 

.99402 

1-8 

12.9591 

13.364 

1-4 

25.9181 

53.456 

3-16 

3.73064 

1.1075 

3-16 

13.1554 

13.772 

3-8 

26.3108 

55.088 

1-4 

3.92699 

1.2272 

1-4 

13.3518 

14.186 

1-2 

26.7035 

56.745 

5-16 

4.12334 

1.3530 

5-16 

13.5481 

14.607 

5-8 

27.0962 

58.426 

3-8 

4.31969 

1.4849 

3-8 

13.7445 

15.033 

3-4 

27.4889 

60.132 

7-16 

4.51604 

1.6230 

7-16 

13.9408 

15.466 

7-8 

27.8816 

61.862 

1-2 

4.71239 

1.7671 

1-2 

14.1372 

15.904 

9 . 

28.2743 

63.617 

9-16 

4.90874 

1.9175 

9-16 

14.3335 

16.349 

1-8 

28.6670 

65.397 

5-8 

5.10509 

2.0739 

5-8 

14.5299 

16.800 

1-4 

29.0597 

67.201 

11-16 

5.20144 

2.2365 

11-16 

14.7262 

17.257 

3-8 

29.4524 

69.029 

3-4 

5.49779 

2.4053 

3 4 

14.9226 

17.721 

1-2 

29.8451 

70.882 

13-16 

5.69414 

2.5802 

13-16 

15.1189 

18.190 

5-8 

30.2378 

72.760 

7-8 

5.89049 

2.7612 

7-8 

15.3153 

18.665 

3-4 

30.6305 

74.662 


Circumference of circle = 3.1416 X diameter. 

Area of circle = 3.1416 X radius squared. 

Area of circle = .7854 X diameter squared.. 


55 













WOODRUFF SYSTEM OF KEYS 


Nfi 

length* 

THICKNESS 
W- L-H 

WIDTH 

DEPTH 

IN SHAFT 

SMEAR 

BASED 

ON # 

5oooo 

PEF?°IN 

SHAFT 

DIAMETERS 

D 

xx 


1 


1-16 

13 

.172 

1568 

5-16 TO 3-8 

2. 

1 

3-32 

.157 

2 350 

7-1 6 TO 9-i 6 

3 

2 

1 —8 


.14 1 

3 132 

5-8 To 3-4 

4 


3-32 

X 

204 

2937 

7-1 6 TO 9-1 6 

3 

5 

1 -8 

.1 87 

3915 

5-8 TO 3*4 

. 6 

8 

5-32 

4 

.172 

4894 

13-1 6 T 0 15-16 

7 

3 

1 -8 

r 

250 

4700 

5-8 to 3-4 

S 

3-32 

O 

.234 

5872 

1316 To U5-I6 

9 

•4 

3-16 

16 

.219 

7050 

1 To |,|-I6 

1 O 


5-32 


.297 

685 0 

7-8 To 15 -I 6 

1 1 

7 

3-16 

3 

.282 

822 1 

1 To 1,3-16 

1 Z 

"5 

7-32 


.266 

9591 

1,1-4 To 1,5-1 6 

A 


1-4 


.250 

1 096 1 

1 , 1-2 To 1,5-8 

i5 


3-1 6 


344 

9375 

1 To 1,3-16 

14 

I 

7-32 

7 

__ # 

.328 

10937 

1,1-4 To 1,74 6 

15 

1 

1-4 

|6 

.312 

1 2500 

1,1-2 To 1.5-8 

B 


5-16 


.28 1 

I562S 

1 To 1,3-16 

16 


3-16 


391 

1 0545 

IJ-I 6 T 0 1,3-16 

17 

,! 

7-32 

31 

.375 

1 2305 

1,1-4 To 1,7-16 

18 

1 8 

1-4 

64 

.359 

i4 be 2 

1,1-2 To 1,3-4 

C 


5-16 


.32 8 

1 7575 

1 To 1,346 

19 


3-16 


454 

11718 

1,1-1 6 T 0 1 ,3~l 6 

ZO 

t 

7-32 


438 

1 367 1 

1 , 1-4 Tb 1,7-16 

2.1 

I 4 

1-4 

00 

.42 2 

15625 

1,1-2 TO 1 , 3-4 

D 

5-16 

64 

.391 

1 9530 

l toI,3-IS 

£ 


3-8 


.360 

23436 

1.7-8 To2, 1-4 

2.2 

1 3 

1-4 

|9 

4 69 

17187 

1,1-2 To 1 , 3-4 

2.3 

'a 

5-16 

32 

438 

2 I4fl^ 

U3rl6To2 

r 


3-8 


407 

2578 1 

1.7-8 TO 2, 1—4 

24 

1 

1 -4 

Ul I 

.515 

1 8 750 

1 , 1-2 TO 1,3-4 

25 

I? 

5-16 

74 

4S4 

23437 

I,I3H6T©2,1-2 

G 


3-8 


453 

28125 

1,3*4 TO 2, 1-4 

26 


3-16 


438 

15910 

I.I-I6TO 1,3-16 

27 

2d 

1-4 

17 

406 

2088 8 

1,1-2 TO 1 , 3-4 

28 

*-8 

5 -IS 

32 

375 

253 1 2 

I. 3-4 -re 2 

29 


3-8 


.344 

29702 

1,7-8 TO 2,1-4 

30 


3- 8 


750 

53870 

2 T02.I-2 

3l 

-z 1 

7-16 

IS 

.719 

6 1 840 

2,3-8 TO 23-4 

32 

3 2 

1 - 2 

l(S> 

.687 

69525 

2,1-2 to 3 ,1-2 

55 


9-16 


656 

7678 1 

34-2 To-4.1-2 

34- 


3-8 


.625 

839'8 

3,1-2 t 0 5 


SELLERS STANDARD 

KEYS 


A 

Size of key. 


Diameter of shaft. 


5 v 3 


1 1 tn 1 1 1 

A 4 10 a t? 


7 V 1 

If a 2 


115 tn Q 3 

A T 6 10 W I6 


9 v 5 


2 A 


1 1 v 3 


2}| to 3 t \ 


1 3 v 1 


[m to m 


if XI 


6 A to 6 t \ 


1 A X 1 i 


fil 5 Q1 5 
°T6 10 °T(T 


Length of key seat 

for couplings H 

X nominal diameter of 

shaft. 


STANDARD TAPERS 



MORSE 



Number of taper 

1 2 

3 4 5 

6 

Inches taper per foot 

.600" .602' 

7 .602" .623 .630 

.626 

Diameter at large end 

.568 .706 

.947 1.249 1.761 

2.510 

Depth of socket taper 

H 2 t \ 

3 A ^ A 5 A 

H 

Diameter large end of socket .475 .700 

.938 1.231 1.748 

2.494 


56 

























BROWN & SHARPE. 


Number of taper 

4 

7 

8 

9 

10 

Inches taper per foot 

. 05 

.05 

.05 

.05 

.05 

Diameter at large end 

.403 

.728 


1.082 

1.315 

Depth of socket taper 

1 5 

1 TiT 

i/ 1 5 

6 

Q 1 7 
d 3 2 

4 

r, 1 

Diameter large end of socket 

1 1 

3¥ 

.720 

.808 

1.060 

1.287 


In all exercises there are certain details of finish which should be 

carefully observed and before submitting drawings to instructor the 

following question should be asked the student by himself and satisfac¬ 
torily answered: 

In pencil work: 

1st. Are border lines, cutting lines and title finished? 

2nd. Are all center lines in, arrow heads and dimensions complete 
and right side up? 

3rd. Are all finish marks, designating letters, material marks and 
number of parts wanted, given? * 

4th. Are all figures and letters vertical and fa" high? 

In tracing work: 

1st, 2nd, 3rd, and 4th as in pencil work. 

5th. Are picture lines approximately fa" thick and other lines 
light and smooth? 

6th. Are-the junctures of lines and curves smooth? 

LETTERING EXERCISE. 

Layout standard 18"X24" sheet and put title stamp in lower right 
hand corner (page 7). Divide the sheet for 11 lines of letters of 
heights 2"—1£"——1"—1"—-j" and five lines fa". Arrange the lines 
of letters at such distances apart as will give good proportion to the sheet. 
Erect vertical line through middle of sheet and space the letters symmet¬ 
rically on either side of it (pages 10, 11, 12). Place on the respective 
lines the following: 

1st. e f h i l. t 2" high, 7 spaces wide, 2 spaces between 
letters. 

2d. a k m n w x y z 1-J-" high, spacing as above. 

3rd. b q J P Q R s & 1|" high, spacing as above. 

4th. extended gothic, 1" high, Y spaces wide, 1 space 
between letters and 4 between words. 

5th i 2 3A S6TB90.1" high, Y spaces wide, 2 spaces 
between figures. 

6th. Your own name, high, l space between letters and 4 be¬ 
tween words. 


EXERCISES. 


7th. to 11th. “ROR AL-L. ORDINARY NOTES, 

DIRECTIONS AND DIMENSIONS, USE 
EXTENDED GOTHIC LETTERS THREE 
THIRTY - S E O O N D S OR AN INCH IN 
HEIGHT, AND WIDTH ABOUT ONE AND 
ONE HALE TI IVIES THE HEIGHT. MAKE 
THE LETTERS FREE HAND WITH A PEN¬ 
CIL NOT SORTER THAN 2 H WITH POUND 
SHARP POINT. GUARD AGAINST AN V 
SLANT. OBSERVE GENERAL PRO¬ 

PORTIONS OR MECHANICAL GOTHICS. 
KEEP SPACES BETWEEN LETTERS 
SMALL AND BETWEEN WORDS GEN¬ 
EROUS.” 

BOLT, NUT AND THREAD EXERCISE 

Lay out standard 18"x24" sheet. Divide the sheet into quarters 
and put title stamp in lower right hand corner (page 7). 

1st. Draw a standard 3"x6" bolt and nut with U. S. Standard threads, 
showing three views (page 16). 

2nd. Draw 3 studs 2"x6", threading each end 2" as follows: 
First stud, ^"pitch R. H. single V thread on upper end and 
pitch L. H. double V thread on lower end. Second stud, T 5 ¥ " pitch R. 
H. single U. S standard (Seller’s) thread on upper end and T 5 g-" pitch 
L. H. double U. S. standard thread on lower end. Third stud, pitch 
R. H. single Powell thread on upper end and V' pitch R. H. triple 
square thread with helical outlines on lower end (pages 12, 13, 14). 

3d. Draw three conventional bolts with hex. nuts, using different 
thread conventions. 1 — 1"X5", 1—f"X4"and l — t\X3". 

4th. Draw 6 types of cap screws of lengths frorq 1^" to 2f" by 
quarters and diameters from §" to by sixteenths (page 52). Draw 

also six types of set screws of lengths from 1 to 1|" by eighths and 
diameters from P'to to ^ f" by sixteenths. Print name of each beneath 
(page 16). 

TABULATION. 

Lay out standard 12"xl8" sheet with title stamp in lower right hand 
corner. Count the longest word or number of figures in each column 
and proportion the width of each column to correspond. Print longest 


58 



EXERCISES. 

heading in its appropriate column and with this as a measure draw line 
for top of column of tabulation. Divide the column into as many spaces 
as there are items in any column, draw lines horizontally and parallel to 
each one, away, draw others. Print in its proper place the longest 
number in each column and through each figure draw light vertical lines 
to locate other figures on. Keep figures of the same unit in the same ver¬ 
tical line. The free hand tabulations in this book will serve for sug¬ 
gestion. 

HAMMER HEAD. 

Lay out 18"x24" standard 
sheet and stamp lower right 
hand corner. First draw in the 
upper half of the sheet the ham¬ 
mer head shown in Figure 89, 
giving three views. Locate 
necessary points and draw cor¬ 
rect outline of flattened face. 
The hammer is tool steel forged approximately to shape with eye 
swaged in and all finished on lathe and shaper. In lower half of 
sheet make three drawings of one end and neck, one with hexagon 
head, another with octagon head and another with square head. Let 
the circumscribing circle in each case be 2-^-" and the neck of diameter 
and radius as shown in figure. 

MARINE TYPE CONNECTING ROD END. 

Lay out standard 18"x24" sheet and stamp in lower right hand 
corner. 

1st. Draw three views of connecting rod 
end, Figure 90, and project necessary points 
to locate curve formed by the intersection of 
the plane and pseudo spherical surface (page 
37). The machine work is lathe, shaper and 
drill press. 

2nd. Design and draw two half journals 
for 2f"x2f" crank pin, to be bolted to the 
connecting rod end. Design the journal box 
to be cast in iron in one piece, babbitted, 
all surfaces finished on lathe, drilled, reamed 
and cut in two on milling machine. 

3rd. Detail special bolt with round head 
boxes to connecting rod. 




59 
























EXERCISES. 


RIGID PILLOW BLOCK. 


Lay out standard 18"x24" sheet, 
stamp lower right hand corner. Lay 
off center lines for plan, front eleva¬ 
tion looking in at the end of the bore 
and right elevation, allowing propor¬ 
tional margins around the views. 
Make a full size working drawing 
with finish marks and all necessary 
information for shop. Project neces¬ 
sary points to locate the curve of 
intersection between screw lug and 
barrel. Take the dimensions from 
Figure 91. 



TWISTED BELTING. 

Lay out standard 18" X 24" sheet with title stamp in lower right hand 
corner. Draw vertical line through middle of sheet. 

In left hand half make a drawing of a quarter twist belt, connecting 
two pulleys 12"X24" and 12"X36" on shafting apart. Show the 36" 
pulley (the driver) in circular view at the bottom of the sheet, rotating 
with the hands of a watch. Arrange pulleys so that belt approaches in 
plans of pulleys. Make to as large a (regular) scale as possible, showing 
plan and elevation. 

In right hand half show the same problem, only with the shafting 
making 30° in plan, instead of 90° In each case show pulleys in out¬ 
line and shafting by center lines 


STEEL COAL HOPPER. 

Lay out standard 18" X 24" sheet with title stamp in lower right hand 
corner. 

Draw a conical coal hopper 6' diameter and 4' deep with 10" pipe 
6' long leading from center of the hopper at an angle of 18° with the 
axis. Develop the cone and the cylinder and detail parts so that sheets 
of number 10 gage 3'X6' will cut to the best advantage allowing for a 
lap of 1J" at joints. Use J" rivets 2" pitch. Detail steel band 
to go outside the rim of the top of the hopper. Give all dimensions and 
notes necessary for shop man to make without having to ask any ques¬ 
tions. 


oo 


















STRAP END CONNECTING ROD 

Lay out standard 18" X 24" sheet with title stamp in lower right 
hand corner. 



G1 






































EXERCISES. 


1st Sheet. Detail all parts of Figure 92 with such views as are nec¬ 
essary to give clear working drawings. Describe machining of each part. 

2nd Sheet. Make assembled drawing from previous sheet showing 
three views. 

TAIL STOCK. 

1st sheet. Lay out standard 18"x24" sheet with title stamp in lower 
right hand corner. 

Detail spindle, center, spindle screw, hand wheel, nut, screw bush¬ 
ing, and clamp screw, shown in Figure 93, full size. 

2nd sheet. Lay out 18"x24" sheet as above. Locate and draw center 
lines and make, to scale of 12"=T', plan, front and side elevations of 
assembled tail stock from previous drawing. Make front elevation in 
partial section to show internal arrangement, and place with hand whee 
at right. Design means of fastening tail stock to lathe bed, also means 
of preventing spindle from turning. 



S-INCH SPEED LATHE HEADSTOCK. 

Lay out standard 18"x24" sheets with stamp in lower right hand 
corner. 

1st sheet. Detail each piece with as few views as necessary to make 
clear working drawings. Dimension carefully and give any other infor¬ 
mation necessary to put work through shops with no questions arising. 
Tabulate in upper left hand corner each piece with designating letter. 
The front and rear ends are cast solid, machined and then cut on one side 
then clamped with fibster screws. The pin holding cone to spindle 
should taper all one way to facilitate construction and removal if neces- 


(52 








EXERCISES. 


sary. Do not employ arrows or thread conventions shown in Fig. 94. 

2nd sheet. Draw center lines for plan, front, right hand and left 
hand views.. Make such sections as are necessary to show construction 
and put in hidden lines only where necessary to avoid confusion. Give 
general dimensions of cone, spindle and base with designating letters for 
each piece. 



GEARED PUMP. 

Lay out standard 18"x24" sheets with stamps in lower right hand 
corners. 

1st sheet. Detail full size all moving parts of pump excepting pul¬ 
leys and gears. Make as few views of each piece as is consistent with 
clearness. Dimension fully and give proper directions as to finish, 
number wanted, material, designation number, etc., whether shown in 
Figures 95 and 96 or not. 

2nd sheet. Detail frame,gears and pulleys 6 in. = 1 ft., giving neces¬ 
sary views, sections, dimensions and notes necessary to send it through 
shops without question. 

3rd sheet. Make assembled drawing and tabulate in upper left hand 
corner a list-of all pieces, screws, oil cups, etc.,with designating letters. 


63 








































EXERCIS ES. 


t 


4 




FIG. 95 


64 














































































EXERCISES. 



C l COLLAR 
l-WANTED 
DNlSH 


J2DP GEAR 3*D 
l WANTED 


TIGHT a.LOOSE PUL-LEYS 
l-EACH WANT ELD 

FIG. 96 


CONIC SECTIONS. 

Lay out standard 18"x24" sheet, with stamp. Divide sheet into 
four equal parts by drawing horizontal and vertical center lines. 

1st. Given a right cone with circular base, altitude 4", diameter of 
base 4". Cut cone with a plane in such a way as to give an ellipse. 
Determine intersection of plane and cone on all three views, and show 
true shape of section a d name it. 

2nd. Given same size of cone as above. Cut with a plane so as to 


% 


65 








































































EXERCISES. 

a 

give a parabola Find intersection on all three views and show true 
shape of section and name it. 

3rd. Given same size of cone as above. Cut with a plane so as to 
give hyperbola. Find intersection on all three views, and show true 
shape of section and name it. 

4th. Given a right cylinder, 5" long, 3" diameter. Draw three 
views and cut cylinder by a plane making angle of 75 with its axis 
Find intersection on all three views, show true shape of section and 
name it. 

INTERSECTION AND DEVELOPMENT OF CONE AND CYLINDER. 

»> 

Given a right cone with circular base 4" in diameter, and altitude 
of 5", placed with its axis perpendicular to the horizontal plane. The 
cone is intersected by a cylinder 2^" in diameter and 4^" long. The 
axis of the cylinder has a direction parallel to both the horizontal and 
vertical planes and passes §" in front of the axis of the cone and 2" above 
its base. 

Lay out standard sheet 18"x24" with stamp. In the upper left hand 
portion of sheet, draw plan, front and side elevations of model as de¬ 
scribed. Show both visible and invisible portions of the line of inter¬ 
section of models on all three views. Develop the convex surfaces of 
cone and cylinder and show the shape of opening in each. 

PROJECTIONS AND REVOLUTIONS. 

1st sheet. Lay out standard 18"x24" sheet with stamp. Divide 
sheet into four equal parts by drawing one horizontal and two vertical 
lines. 

1st. Draw plan, front and side elevations of.model placed with axis 
vertical and one edge of base parallel to vertical plane. Name the three 
views and the three planes. 

2nd. Revolve model about VPN 30 to the right and draw three 
views. Name views and planes. 

3rd. From position as left in 2nd, revolve model 30 about H P N 
and draw three views. Name views and planes. 

4th. Draw plan, front and side elevations of model placed with 
axis inclined 60 ' to the horizontal plane and parallel to the vertical plane. 
Name views, and planes. 

2nd sheet. Lay out standard 18"x24" sheet with stamp. Divide 
sheet into four equal parts by drawing horizontal and vertical center 
lines. 

it 

1st. Draw plan, front and side elevations of model placed with axis 
of hole vertical and one edge of base parallel to vertical plane. Name . 
views and planes. 


Lof C. 


66 



EXERCISES. 

2nd. Revolve model about H P N 15 to the right and draw three 
views. Name views, and planes. 

3rd. Revolve model about P P N 30° toward the front. Name 
views and planes. 

4th. Revolve model about H P N until the edge which was origin¬ 
ally parallel to the vertical plane becomes parallel to it again. Name 
views and planes. 

CONIC SECTIONS BY MATHEMATICAL RELATIONSHIP. 

Lay out standard 18"x24" sheet with stamp in lower right hand cor¬ 
ner. Divide space inside borders into 16 spaces 4f"x5f. In 6 spaces 
locate points of one quadrant of ellipses 2x"4f" by different methods and 
approximate complete curves by arc method. 

On two of the ellipses so made draw tangents and normals to some 
point by different methods. 

In 5 spaces draw parabolas by different methods with a tangent to 

one. 

In 5 spaces draw hyperbolas by different methods and with tangents 
to two by different methods. 

In each case denote points found in one quadrant by small circles 
drawn with bow pencil before approximating curves by arc method. 
Name each figure and give name of method. 

MISCELLANEOUS CURVES. 

Lay out standard 18"x24"sheet with stamp in lower right hand corner. 
Divide space inside border lines with faint lines (to be erased) into 6 
spaces 7f"x8f". 

In one space draw with very light lines a triangle with base 2f" 
long and two sides each If" long. With one end of the base as a pole, 
pass a logarithamic spiral through the other two corners. Determine 
angle a tangent makes with radial line. 

In one space draw a helix with diameter of cylinder 4" and rise of 
2" per revolution. 

In one space lay off from a base line angles of 13 -20', 28- 10' and 
52"-40*. 

In one space locate points of an involute with base circle of 4" 
radius. Make curve about one inch long. 

In one space draw hypocycloidal and epicycloidal curves on a pitch 
circle of 6" radius and with generating circles of 3" diameter. 

In one space draw, by different method from the preceding, on a pitch 
circle of 3" radius, a hypocycloid with generating circle of If" radius and 
an epicycloid with generating circle of 1" radius. 

Denote points in involute and cycloids by small circles, and do not 
draw in the curves. 

Name each curve and the method employed. 


«7 


* V 


EXERCISES. 

TRILOBE CYCLOIDAL PUMP IMPELLER. 

* 

Lay out standard 18"x24" sheet with title stamp in lower right hand 
corner. Divide sheet by vertical line through middle. 

In one part constuct on a 6" pitch circle an epicycloid with 1" gen¬ 
erating circle which shall return again to pitch circle. From the point 
of return start a hypocycloid with 1" generating circle. Denote points 
formed with small circles and approximate curves with arcs There will 
be three lobes and three depressions. 

On the other part of the sheet repeat the problem by another method. 
Test curves by zinc templet in custody of instructor. 

CEAR PINION AND RACK. 

Lay out standard 18"x24" sheet with title stamp in lower right hand 
corner. 

Construct pitch circles of 8" radius and 16" tangent to each other 
on a line of centers. Tangent to the 8" radius draw a line for the pitch 
line of a rack. Lay out four teeth on gear and pinion and four teeth on 
pinion and rack, all of | D P. Denote line of action and arcs of contact. 

BEVEL GEARS. 

Lay out standard 18" X 24" sheet with title stamp in lower right hand 
corner. Divide the lower sheet vertically in parts 16£" X 14" and 16V' X 

8 £ " 

In the larger part lay out two bevel gears 12 and 24 teeth, 3 D P. 
Develop tangent cones and lay out three teeth on each gear by Grant's 
method. Dimension for working drawing. See Figures 80, 86 and 87. 

In the smaller space make pictorial drawing of smaller gear with 
three complete teeth showing views of blank looking at edge perpen¬ 
dicularly to shaft and at inside and outside parallel to shaft. 

GEARINC EQUIVALENTS. 

1st Sheet—Lay out standard 12"x 18" sheet with title stamp in 
lower right-hand corner. Calculate and tabulate the equivalents in D P 
units, indicated below, using B. and S. proportions Print out the full 
name as well as the symbol, putting the symbols in one column and the 
names in another. Describe in words same operations as formulae in a 
separate column. Arrange table in such a clear manner that, having 
given certain data and wishing it in other terms, one can find it with per¬ 
fect ease 

To get C P having given: 1st, D P; 2nd. D and N; 3rd, A D and N 

To get D P having given: 1st, C P; 2nd, N and D; 3rd, A D and N. 

To get D having given: 1st, N and C P, 2nd, D and C P, 3rd, N 

and A M. 

To get N having given: 1st, D and C P. 

To get T having given: 1st, C P; 2nd, D P. 

To get A M having given: 1st, C P; 2nd, D P; 3rd, A M. 

To get R D having given: 1st, C P and N; 2nd, D P and N. 

To get A M -f D M having given: 1st, C P; 2nd, D P. 

To get A M | DM -f C having given: 1st, C P; 2nd, D P. 

To get C having given: 1st, C P; 2nd, D P; 3rd, T. 

2nd Sheet—Same as first, excepting that equivalents be in C P 
units. 


68 


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A D VERTISEMENTS. 



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» 


70 



















■ 

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A D VERTISEMENTS. 


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DRAWING AND SURVEYING INSTRUMENTS 

POCKET CUTLERY = 

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Manufacturers and Importers of 

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Try a Bottle of Post’s Superior Waterproof Drawing Ink. 


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71 














































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4 

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